Seriously, though, Depot A is absolutely demolishing me. [...] Any tips or advice would be really appreciated.
What part of Depot A? There's the "Pre-Depot" part which is the first maps you have access to, with Junkyard Muties, a few dogs, some Mutated Dogs, and a few turrets; then, the "Depot proper" which is what you get access to once you go through the locked gate, with full-on Mutants and Mutated Dogs. If you're having trouble with the first part, then it's entirely how you're playing the psi character. For example, you say you're always going last - you need to be manually starting combat, so you go first; start doing that, and you'll see a marked improvement right there. On the other hand, if you're having trouble with the second part, it's possible you need to go pick up one or two more levels. There's no reason at all that you should ever be under level 8 when starting the Depot proper; if you're playing Classic XP, you can easily manage level 12 on DOMINATING, or even 13-14 if you really do absolutely everything first.
Pick the Pyromaniac feat at level 8 and you'll be able to crowd control the Mutated Dogs when fighting the Mutants; Mutants are immune to fear of fire but the dogs aren't and you can fearbounce them pretty reliably. Between your four-turn Force Field (because you took Force User at level 4, right?) and your Premeditated Pyrokinesis (because you took Premeditation at level 6, right?) you should be able to throw down a couple caltrops (if you made/bought a stealth set, maybe even a few traps too) and get everyone in the entire map to bunch up at the edge of your force field, then pop Adrenaline and throw a grenade, toss out a fireball, and lob out an Electrokinesis and a TK Punch to get some stuns, then hop out of LOS to end your turn. Next round, do whatever you gotta do to survive - hopefully, it'll involve killing a few more enemies, then maybe finishing with a Morphine Shot if you see that you're going to have to facetank a Mutant. With a LTI, you'll then be able to hit a second psi booster for the fight and mop up. Absolute worst case, you do stuff wrong and lose control of the fight - zone out, heal up, let all your cooldowns refresh, and go back in. It won't be as controllable the second time walking in, but at full health and with consumables and all your resources topped off, you should be able to survive one turn (again, you can pop Morphine if needed to survive that first messy turn walking into combat) and lay down the law. By which I mean murder everything.