Ending a sentence with a contraction is entirely valid in normal English.
I tried to force myself to eat the last bite of cheesecake, but I just couldn't.
Oh, go on. I'll eat this whole chocolate bar, even though I know I shouldn't.
No, really. I mustn't.
Really. Don't do it. Just don't.
Put a spider in her bed when she's sleeping? You wouldn't!
You two are going out, be we aren't.
Your ice-cream is tasty, but this one isn't.
You want to go to the mall? Yes, let's!
I can't remember sending that email, but I must've.
I didn't do it, but I could've.
The time now is eight o'clock.
How may I help you ma'am?
Greets, y'all! (warning: not standard English)
In fact, in the above, use of the non-contracted forms instead of the contracted-forms sounds stilted, although your point will still get across.
Note that there are some contractions where, as Bill points out in the comments below, one would not normally make use of them at the end of a sentence:
She's not going home for Christmas but I am (not I'm).
She's not going home for Christmas but we are (not we're).
You hadn't eaten a chocolate pudding, but I had (not I'd)
That experiment isn't awesome this project is (not project's)
That idea won't work, but this will (not this'll).