I cannot figure out what is the purpose of turn shove with staight on paired board? I would probably just cbet flop and fire another bet ott on hit. Folding to any raise ofc.

As played probably x-calling turn once again is the play. But i have no idea where it leads.

I agree that getting it in on the turn doesn't seem to do anything for you, your probably far behind his calling range.

Given his bet sizing otf (seems weak to me), I'd rather lead ott and get value otr on safe cards

On TT9r, I think betting a more polarized range and checking a lot of the middling stuff is going to be better.  The more your hand needs protecting from villain's folding range the more you should be betting those hands.  Additionally, you need to be aggressively protecting your checks with lots of calls and raises as a lot of his flop bluffs often have significant equity. 

This hand seems to play better as a check/call as we don't want to bet and strengthen his flop calling range(which makes our hand less valuable when we improve to a straight.  However when we check and keep his range wide, our straight is often strong enough to get 2 streets of value) and we have a hand that can often check/call two streets. 

As played, the turn is a fairly clear check/call.  I don't see any reason to raise as his weaker holdings have 20% or less equity vs our hand (so protection doesn't matter) and by raising he folds out worse value hands and gets it in with hands we're dead against.  Additionally, it's a turn card he's likely barreling pretty often as it's pretty bad for the overpair portion of our range, so we're likely to gain at least a turn bet from his bluffs. 

I would be tempted to check and call a river bet with this combo on most rivers.