NL 5 - Pocket 10's

Flop is not a mandatory cbet. But bet smaller. With your sizing you are kind of obligated to shove against a check/raise.
Turn is interesting. I expect a good player to lead here after you bet bigger OTF. Your flop sizing is indicative of a hand that needs protection and that's why. Besides, you are playing 64bb effective (villain's stack) OTT on a 50bb pot.
The ace is the only card he can profitably bluff. In fact I think he should bet really wide here (basically his whole range with such a short stack). And you should probably fold here TT against a decent bet.
T9s, 76s, 97s would be better calls for instance because you wouldn't be blocking so many bluffs (KT, JT, T9...).
This is not an easy spot even when he checks. If you bet and he shoves you have a decision. If you bet and he calls, he'll be committed for the river.
I think I'd bet small OTT and try to take the pot to showdown. There are still some worse hands that might calls you (pairs, flushdraws). If he shoves I guess I would fold.

I'm the worst player ever. I would've cbeted 1/2 flop and turn, probably folding to a check/raise OTT.

Considering that villain is a short stacker and therefore a potential recreational player, I might prefer betting flop and turn. However, I agree with the guys above about the bet size being on the larger side OTF.