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Honestly the primary problem with guns is that people overthink and overdesign them mostly due to the design philosophy of weapons.
Every D&D weapon is designed, in large part, mindful of the myths of the weapon in mind. Swords are a noble's weapon meant for fighting other people, so there are like a billion magical swords just lying around in every official adventure (after all, only nobles could afford enchanting that many weapons). Axes are the savage weapons of brutes, and therefore barbarians have abilities that are specifically designed to work with greataxes but not greatswords. Staffs are the weapon of the wizened, so they also function as a magical focus for wizards. Daggers work with sneak attacks, but your fists don't, because assassins use daggers in the night, while martial artist make a big show about punching you in the face. And this also applies guns. Guns are thought of as weapons of little skill needed and immense power output, to the point where they violate the rules of all other kinds of combat, obsoleting everything else.
What happens when people consciously or subconsciously work this idea into the rules of guns, is that people put too much power in the gun itself, and not enough in the character wielding the gun. Which is totally backwards compared to every other weapon in the game. Swords and bows are mundane and lethal, yes, but even mundane martial characters do impossible things with them, like interrupt someone else's turn and attack 8+ times in a fraction of a second (hi to you, Samurai) or boost the power of a single attack to do over 4X damage (Pick any Rogue). So naturally, when you make a weapon that is supposed to break the rules of the weapon subsystem, it breaks the rules of the weapon subsystem. They just don't jive, because they aren't supposed to jive.
So how are you supposed to put a square shaped weapon into a circle based hole? You don't. Instead you look for a subsystem that fits better. You want an object, that deals significant burst damage, can't be significantly boosted by the character using it, and has limited uses?
You want a magic wand. One shot takes one charge. "Reloading" the wand (with whatever frame of time you want)gives you another charge. You could even have special guns with multiple charges (like double barrel guns). You can set the damage to be however high you want. You can set the default accuracy to whatever you want. You can customize the damage type, area of effect, exactly how much noise and flash and even smoke they give off. Basically everything is fair game once you stop designing the gun as part of the weapon system.
Or, you can just make them slightly tweaked copies of crossbows that make a loud noise. If you are more interested in making the character using the gun more important than the gun itself.
I think this is the best answer: use feats, class abilities and other mechanics built into the wielder to differentiate guns and make them fun and cool. And, for the love of all that is holy, do not try and model real guns from any era. 5E doesn't do that with anything else, why would you do it with guns.
What might be fun is a gun-centric subclass for each main martial class: fights have a traditional gunslinger, rogues have the silencer assassin, rangers have the long arm rifleman/sniper, barbarians go full rambo, and monks get John Wick gun-fu.
I think the single shot and drop feels right for a pirate age of sail feel. Particularly if you are looking to keep modern tech down and not have machine guns and rapid load six shooters etc..
I think it kinda depends on the 'fantasy' you want to portray.
If I want guns in a setting, I want them to be big deals. In 5E, weapons are mostly flavor, because what is exactly king isn't what you are using (outside of magic items), but how many times you are using it. Action Economy + HP bloat are the big reason why, but also why shoot somebody when you can banish them?
Guns are scary in systems where HP is low and combat can be over in the blink of an eye. If you play Deadlands, you can realistically have your head blown off from a single attack from a sniper rifle. Or a full-auto weapon in Call of Cthulhu (once you get past the clunky full-auto rules) can decimate a warehouse full of cultists in a single round. To me, that's what firearm combat should feel like.
this is definitely an interesting solution, and i think it could work. you could even, theoretically, use it to help a bit with the martial/caster divide, by letting martials pick up various firearms as powers as they level up. it'd be kind of weird, though, and much more in line with 4e game design then 5e, but it's a route you could go.
from a verisimilitude perspective however, it's a bit bizarre. in reality, it takes about as long to load a (smoothbore) muzzleloader as it does to load a heavy crossbow, and in 5e at least, that happens between turns (or between attacks with crossbow expert - which is personally my biggest problem with that feat!). then there's also the question of what happens when a PC goes full pirate and decides to roll up with a full bandolier of pistols. do you just let that happen, or do you arbitrarily cap how many guns you can fire before a short rest?
there's also the problem inherent to this approach, which is that it completely negates the idea of a firearm as a primary weapon. you cannot have a character who mains an arquebus like another character mains a greatsword or longbow or heavy crossbow, because they can only use that arquebus, at most, once per fight (more likely once every other fight), and then they have to swap to something else. that entire aesthetic simply cannot exist with this method. which, depending on the campaign, could be fine...but i don't really think it's great as a universal solution.
the solution i've pondered for a while is to give firearms a second damage die and make their damage dice explode, but don't allow ability modifiers to be added to their damage. that does a few things:
1. it reduces the gap between skilled and unskilled firearm users (which is a big reason they got as popular as they did to begin with) by removing the ability modifer to damage.
2. it keeps firearms about as capable as other ranged weapons by throwing in the second damage die. a 2d4 firearm is about as damaging as a hand crossbow wielded by someone with a +3 dexterity modifier, for instance.
3. it somewhat displays the absolute horror show that is terminal ballistics - both by removing the ability modifier to damage (e.g. a bullet overpenetrating a target and causing minimal damage) and by making the damage dice explode (e.g. a bullet not only expanding within the target, but also possibly bouncing off of bones and causing even more damage).
1 and 3 help firearms feel unique and (at least somewhat) true to life, while 2 prevents firearms from becoming over-or-underpowered, since they're pretty much always within the expected band of ranged damage anyway (well, aside from the exploding dice, but you could remove that if it really worries you). in turn, because the firearms aren't overpowered, you can sling them around as a primary weapon instead of just as an encounter power. in theory, anyway.
I think it kinda depends on the 'fantasy' you want to portray.
If I want guns in a setting, I want them to be big deals. In 5E, weapons are mostly flavor, because what is exactly king isn't what you are using (outside of magic items), but how many times you are using it. Action Economy + HP bloat are the big reason why, but also why shoot somebody when you can banish them?
Guns are scary in systems where HP is low and combat can be over in the blink of an eye. If you play Deadlands, you can realistically have your head blown off from a single attack from a sniper rifle. Or a full-auto weapon in Call of Cthulhu (once you get past the clunky full-auto rules) can decimate a warehouse full of cultists in a single round. To me, that's what firearm combat should feel like.
Yeah that is the rub of firearms there are many takes. Especially the "if a katana is doing 2D20 damage then surely a gun should do 4D20" mindset. I remember a PF1 game where I usually banned the gunslinger because it was broken AF, but wanted to include it in Iron Gods AP because a lot of cool laser weapons get found. Kid that plays gunslinger decides to go black powder musket...
I think ideally you need to make the rules modular so that you can have one and done, slightly better than crossbows, and BIG DEAL game settings. A lever GMs can control and players can get on the same page where their interest lies.
I don't think the problem is with guns in D&D, but with wanting to have both guns and middle-age weapons, and balance them with each other. Firearms are simply vastly superior than old historical weapons.
Not by much, swords, lances and spears (in the form of muskets with bayonets) were important military weapons well into the 19th century, and battles were often decided in melee. The real game changer was gunpowder artillery, not personal firearms.
Like…do you have different draw weights and materials and constructions of bows, as well?
When someone has a gun in my game, it’s just another weapon. It isn’t deadlier, it’s just a simple weapon so more likely to be in the hands of conscripts and randos.
mindset. I remember a PF1 game where I usually banned the gunslinger because it was broken AF, but wanted to include it in Iron Gods AP because a lot of cool laser weapons get found. Kid that plays gunslinger decides to go black powder musket... 
