Thank you all for the advice and good thoughts! Baby (decided to nickname him so I don't give him a real name since I can't keep him) made it through the night! I was half-convinced that I'd lift the blanket this morning and find him dead, but then he raised his head and blinked at me like "Oh, is it morning?"
He's still having very runny, very green droppings, and has trouble moving around. When I lifted the blanket, he was laying with at least one foot out to the side a bit, not under him, so it looked a bit awkward. But on a positive note, he drank a few more mouthfuls from a spoon, and ate some seed out of my hand! He's cuddling on my chest now. Our appointment is for 2:20 this afternoon, we'll be leaving in a few hours. I'm so glad I brought him home with me for the night, and that I insisted on the appointment... I'll let you guys know tonight how it goes and what I find out. Thanks again SO much for the help, I was so excited to find a budgie forum like the one I go to for my hedgehog!
Edit - Here's a picture I took of him cuddling in my hand too.