I feel like saying this is the kind of anxiety you should face and overcome. I mean, it's not that bad to have a fellow student think your art is bad? Especially not when you aren't an art student. But that's just my unfair reasoning talking.

Having been in similar positions on different subjects, I understand it's not that easy.

I feel that you should think this is a fine way to be out of your comfort zone, but I guess it's more of a confrontation than you'd wish.
I guess that the best answer for me to give is to just put it to you straight that the anxiety you are suffering is not justified. You must have been reenforcing this in your mind in some way. So I feel I have to remind yourself that this is all in your mind. Your brain is doing this to yourself. Yes, it's not a choice to experience this or not, but maybe you can stop the pattern of reenforcing this belief that it is worth it to have anxiety about this.

You mention mental health issues, so I maybe your own judgment of your drawing isn't quite objective and your drawing is much better than you give credit for. And even if you are the worst; so what. You probably get graded on how much you improve. And you aren't a fine art major. And if you were, you would want to know you are terrible at drawing, if that is true, and accept that, and consider what that means for your degree. What are you truly afraid of?

Point being, it's simple for others to say this shouldn't be as big a deal as you make of it, and it is true, but that in itself won't solve your problem. In the end, you yourself need to find a way to make this much smaller of a deal, and then suffer through it and come out stronger.