I'm a bit obsessed about this class. I know that a hexblade multiclass can be a very strong build, but I really would like to play it single class. But... I have issues.
The Hexblade seems to be, on the surface, the classical "Gish" - the fighter/mage that blends swordplay and sorcery to win the day.
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aaaaaand with the hexblade it doesn't seem to quite work,
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I guess you could build an EB based hexblade... but that's more of a hexcheese really). Furthermore, it is easier to build them with a few more utility invocations.
So... what to do without multiclassing?
I have three things for you. (I know it is a wall of text but I assure you it does contain options and serious feedback. I am not concise in my attempt to cover it all but I know there is chance no on will read it. Sorry.)
1. Your last thought is correct, the
Hexblade is actually better as a pact of tome or pact of chain eldritch blast build. Pact of the the Blade looks good for Hexblade but it is actually a trap. Some of the features are redundant to pact of the blade. Only the second half of Hex Worrier applies to a melee weapon and that is useful for attacks of opportunity when used for eldritch blast builds. More to the point, medium armor, shields, and the shield spell do not provide adequate defense to be a "gish" nor do they allow for the flexibility you expect from a warlock since your poring all your invocations into trying to make it work. The Eldritch blast Hexblade gets the defensive advantage of range which is often huge since GMs tend to favor and perhaps over use melee thugs and creatures and you only need one spell and one invocation to bring it online making the rest of your design some the to play with and enjoy.
2. Its often thought of as cheese, but it really is a solid way to play a warlock. Hiding in
Darkness with Devils sight, gives advantage on attacks for offense and Opponents disadvantage on attacks against you for defense based on "Unseen Attackers and Targets" PHB p194-195). Wizards double downed on this with level 4 warlock only spell Shadow of Moil from Xanathar's Guide to Everything which is and improved version of this without the requirement for Devil's sight to free up an invocation. Using your first spell slot to activate this in combat and your second as a backup incase you are hit and fail your concentration save. This means Hexblades spell list is largely a trap with the exception for
the Blink + hex option. Shield Spell especially, since your better off taking hits and multiple saves then recasting darkness if fail, than you are casting shield loosing your second spell slot immediately and no backup out the gate meaning a hit the second round of battle could leave you defenseless. There are a few exceptions with extreme high damage you can't save but you generally don't know the damage before you determine hit, but if you just saw an ally hammered you might have a guess.
3.
I am on board with making a "Gish" warlock but I ran across the same issues. I ended up making a
homebrew to get a subclass that would fix the issue I had. Part of the problem is that warlocks are Multi-attribute dependent and need access heavy armor proficiency for Ring Mail to mitigate that, every warlock subclass to date has a defensive ability at level 6 because they are the only caster class that can't use spells reliably for defense. If you cast counter spell and shield once each your done as caster so they are traps. Magic initiate warlock is a huge help and so a Rod of the Pact Keeper together doubling your spell slots. The Darkness and shadow of moil both make you unseen which help instead of counterspell since many spells require "a target you can see" and shield through enemies having disadvantage versus targets they can't see. The same with Hex + Blink. The problem is that really ties down your flexibly. As you said, in concept your looking at a flexible fighter who uses a variety of spells in combat to enhance and have fun but the level 6 defense abilities don't have but one charge up front, don't recover on short rests, and/or don't scale for greater defense at higher levels the way other casters getting higher spell slots and more defensive spell options can. I suggest, asking your GM to adapt a 6th level warlock from anther patron with at least 2 charges at lvl 6, +1 at 11 and 17. Then make at recharge at least one on short rest so that if your GM does put put your party on the spot with mutiiple short rests and small encounters you can actually do as its expected for warlocks to do.
... Adding to that last point, if you do multi-class to make your Gish work with Ring mail, Cha and Constitution, I recommend the
Fighter subclass Battle master, because the Superiority Dice also restore on short rest. Then assuming your GM actually uses short rests and multiple encounter days some times you will have a bit more to help you be the team anchor when your party expect you to with
Evasive Footwork and
Parry.