Alright, you know the deal, it's a really cool game and in general it feels great and I'm looking forward to seeing more, very impressive, yadda yadda yadda. Let's get right to business and give proper feedback since you're looking for opinions on the movement, which I've listed in severity:
Dragon Tier Level Issues-
*I find it incredibly unintuitive that wall jumps put you into a committed uncontrollable jump arc, like something out of NES Castlevania. Feels very inconsistent and confusing since in general the rest of your jumps have very generous mid-air control, and indeed if you use a double-jump after a wall-jump you get control back, so why is the wall jump so rigid? I don't mind if wall-jumps give you a powerful directional impulse that you can only slightly adjust against, but being completely rigidly stuck in an arc without being able to influence with directions at all feels awful.
*Had a few times where I swear I jumped right at the edge of a platform, only for it to either eat my input entirely and I fell like a log, or for it to use my double-jump instead. Feel like some coyote-time input forgiveness might be in order.
*If you jump off a wall and hold back towards the wall and double-jump, you will as expected double-jump back towards the wall. However, if you jump off a wall and hold back towards the wall and dive, you will confusingly dive away from the wall, completely disrespecting your directional command.
Tiger Tier Level Issues-
*I found it confusing that I couldn't Mega Man X/Meat Boy jump up a wall by wall-jumping and holding towards the wall: instead I have to use a double-jump after the wall-jump to change my direction to go back towards the wall and proceed up. Not too bad since it felt kinda good to get skilled at remembering it, but still, very awkward and against common practice. If the more primary issue with wall-jumps above were fixed, I wouldn't mind if you still couldn't climb with just wall-jumps and you had to use double-jumps to gain height.
*Similar to wall-jumps committing you to an hard-coded arc, if felt a little bit bad that dives also hard-commit you into an arc without being able to influence it slightly. It doesn't feel as bad as wall-jumps and kinda makes more sense, but still, a bit of influence would feel nice (but I deem this a lower threat since it feels more acceptable).
Wolf Tier Level Issues-
*Felt like the wall stickiness was too much: I don't mind the stickiness since it allows people to more easily hold away from the wall and pressing jump to wall-jump without falling off of it too quickly and using their double-jump instead, but in general it felt hard to disengage from a wall when you're trying to rush through a level and accidently graze one, and also it felt like you just slide too slowly: perhaps you should slide down at a fixed rate instead of this awkward increasing speed.
*Wish when you're sticking to a wall it would have some puffs of smoke or a nice sliding sound effect: feels very bland and static compared to all of the other juicy effects for jumping and the like.
*Feels like the run doesn't do much, takes too long to start up, and might just be unnecessary? The basic movement already feels fast and fun enough. Maybe if running gave you a faster initial boost and a satisfying puff of smoke instead of slowly accelerating (like in Kirby, I think?): boosts are more fun than runs.