What should I do?
I'm tired but I can't go to sleep because then I will talk in my sleep, and it's really embarrassing
You can't go without sleep just because you find it embarrassing at the fact you talk in your sleep. Just go to bed, who cares.
Maybe duct tape your mouth while you sleep
In this case, you should excuse yourself to somewhere private. Then, you can choose to sleep there or take a rest as you wait for others to fall asleep. Alternatively, you can try getting a separate room or place if possible.
I love talking in my sleep! =D