A couple of things come to mind here. 99% of songwriting/ being a musician is self doubt (at least in my experience!). Ok, seriously, that's an overstatement, but I think it's true that the way you hear your own songs isn't the way others hear them.
First rule of doing anything. There will ALWAYS be someone better than you. If you let it stop you, you're doing it wrong. Because there will NEVER be anyone who does your thing the way you do it, and that has immense value.
If I were in your shoes, I'd choose what I think is the best of the songs you've written, make sure you can perform it well enough, and just have at it when it's your turn in class. While I understand it's really hard to perform when you think you're not up to standard, maybe having the feedback from your tutor and classmates will give you the boost you need to take your songwriting to the next level.
We all had to start somewhere. But sitting down trying to write to some perceived notion of what's acceptable in a classroom setting probably isn't helpful right now. If you're going to write a song, try and make the reasons for writing it be that you want to write it. Or that you have something to say, something to express, or whatever.
Finally, if the pressure of delivering music to order is making you unhappy, maybe you should consider the role music plays in your life. Music should be fun, and if it's not, maybe it's more suited as a hobby for you? That doesn't mean letting go of the ambition to play/ write to a professional standard, it just means taking the pressure off and doing it because you want to.
Anyway, I hope some of that helps. And good luck with your class performance; I bet it will be better received than you think.