Yeh I think going for a bet on the turn is going to be better than the river. There are oodles of hands villain can have here that can call a bet. HE has some worse Tx and a bunch of potential draws. The Tx hands will likely call a river bet so EV won't change much there but betting the turn will charge all of his draw hands which you can only get value on the river from if they decide to bluff at it.

Also, when we are shorthanded, is going for a bet on the river out of question here? I'm talking about if a blank peels off. If villain just flat calls and checks when a random low card falls, we should have the best hand almost always. Can we go for some value vs his worse Tx hands here?

With the Kx kicker I think you're doing great against villain's range and I would bet. I would be checking back many Tx though, definitely T8 and below and I'm not entirely sure about JT.

Villain probably has a fair amount of worse Tx in his range (people seem to like to flat offsuit broadways in the sb) and I would expect AT to be 3betting pre at least some % of the time. 9x in his range should be relatively low since I assume you're not getting called in the sb with too many offsuit combos and your Tc blocker does a bit of work too.

I tend to always bet these spots if I'm in position. Even if it's not a +EV bet in a vacuum you do protect yourself against some awkward river spots which is worth something.

With a decent kicker and the unlikelyhood of villain not c betting an overpair or 9x here it probably is a decent value bet anyway.