To devs with at least a few years of experience that might not necessarily think of themselves as amazing: Do you still constantly get stumped at work? How has how you handle that situation changed since you first started? Do you still get stressed out?

I have a little more than a year of experience now, and although I've learned a lot, I still feel like I have no idea what I'm doing..

Totally. Over 15 years. It's one of the plus sides of it - there's always a challenge just around the corner. Something to test your logic and problem solving skills. Yes, sometimes you want to scream and cry and run away because it's stressful trying to solve something difficult to a deadline.

And be careful who you compare yourself to. There are developers all along the scale of ability - don't sell yourself short because you think there are others much better than you.

I rarely get stumped, but being stressed out is normal when you're a consultant at a company that practices deadline driven development and estimates are overruled by non technical managers,

Being frustrated is also normal when you're forced to work with shitty tools because tech decisions were made and enforced by non technical managers based on how easy/hard they thought it would be to recruit people with experience using them.

Part of this is your mindset.
Stumped = "Oh dear god I dont know this! Panic!"
Don't let your thoughts end at "I don't know what do."

Reframe it. A problem well defined is half solved.

First:
"What piece of this problem do I need to reconsider?"
Then:
"What resources should I consult to clarify the problem?"
Then:
"What resources should I consult to start generating ideas for solutions?"

It goes from being a roadblock to a learning experience.

Sure. Go for a walk. Or a beer. Or... well I'll just stop there. Context switch.