I find myself agreeing with Leatherhead and Acerak about quite a few of the details about these classes. To get it out of the way, both classes are swimming in flavor, and flavor-wise I want them right now.
Mechanics.....
I think it is important to note that the main power of the Monk 3rd level is coming from that Breath of the Dragon. The ability to "for free" turn one of your attacks (let us say at 5th level a 1d8+4) into an AOE of 2d6? That is not bad. One thing I am curious about is on the potential to burn two uses to use it twice in the same turn. I don't think it would be too broken, but is would lay down some wide hurt.
Compared to that the elemental damage, draconic and re-roll are just sort of ribbons. Though I think I would have their monk weapons be able to deal the damage by 6th level.
Wings Unfurled? Yeah, just improve Step of the Wind to give it flight. In practice, that is what is going to happen, since needing flight more than three times a day is going to be rare. Honestly, I'd almost want to add something else here, but I can resist.
The aura is... thematically neat, but mechanically too weak. Most of my allies likely won't want to use their reaction, plus it only works if they get hit in melee or very close range, because the attacker has to be in the aura as well. Too easy for archers or even melee people who are 30 ft away from you to circumvent this, even if your ally is willing to use their reaction, and even if 1d8 was worth it.
Though, counter-point, group wide resistance to a common element is potentially very powerful. Everyone in 30 ft getting cold or fire resistance can turn a fight by itself. So maybe it is best to think of this as a buff spell, with a minor counter.
The ascendant Aspect.... I feel it is lacking. Sure, blindsight is fun, but everything else is just adding improvements to what you can already do. Now, granted, those are some nice improvements. Your AOE doing "burn" damage every turn to everything? Not bad, especially if you can use it twice and stack those, that could be very devastating. But the 4d10 from the daily aura is a little less impressive, especially since if you are using it for the resistance, at those levels most things dealing elemental damage are resistant or immune to it, making that a real hard set of damage to land.
Though, countering myself again, it doesn't say it has to be the same damage. So, you could set fire resistance and deal cold damage.
And now... the Dracowarden
Man I want to love these mechanics more.
The gift is a ribbon, and a decent on. The lack of spells hurts a lot. But the companion is just kind of bad.
First off, lasting only two hours is a severe limitation. You are going to end up spending at least one spell slot to get it back. Then, it lacks flight to start with, so it is basically just a scaley cat in terms of infiltration or utility. No stealth bonus, so that is going to be hard to do. So, this is purely combat pet, and it doesn't stack up.
First turn, (if not pre-summoned) is summon and bonus action have it attack for 1d6+2.
If it is already summoned then you can attack for, let's say 1d8+3+1d6+2 which is 13? Let's be kind, and say that you get that extra 1d6 from infusing the essence into an attack. So, 16.5.
Dual Wielding alone gets us 2d6+6 which is 13. If I am a hunter ranger I could get colussus slayer for 1d8 extra, bringing me to 17.5. If I could spend a spell slot and get hunter's mark, then by round two I've got 4d6+1d8+6 for 24.5.
And the Familiar never catches up in my opinion.
And like was said, everything in this subclass relies on your familiar being out, and it's hp is decent, starting at 20 and increasing by 5 every level isn't something to scoff at, but the AC isn't great. mediocre really.
And, let's look at the absolute best thing the Drake can do in combat. At 15th level you can use your bonus action to command it to attack for 3d6 damage. That is the max, at 17.5. While, Dual-wielding as a Hunter can get you 2d8+5 which is 14... by as early as 4th level if you roll stats and get a high dex. 13 guaranteed by 8th level no matter what.
The Fire Breath is neat, I won't deny it. I like that ability, but all the rest of it needs to be looked at again, because it is being billed as an offensive option, and it simply does not stack up at all.