I'll just tell you how I usually handle situations like this. Against 3 or more opponents I would just check fold the flop. Against two or less opponents you cant give up that easily IMO.

So back to your situation with TT on an AQx board against 2 opponents. First I would bet the flop. I look at this flop bet as kind of a desperation bet that could easily be my final investment in this hand depending on what happens. If both opponents call, its time to check fold the turn. If one opponent calls, and I have no read on the opponent I will usually bet the turn hoping he called me with a gutshot draw on the flop. There is also a small chance the villain may fold Qx or a hand like JJ, which also adds some value to betting the turn. If the villain calls my turn bet, I will give him credit for a better hand and I will plan on check folding the river. Although there are some hands on the river that you can still beat like any broken flush draw/miss gutshot draw, I believe you wont snap a bluff enough in this spot to make up for all the bets you lose paying off with the second best hands. So with no reads I check/fold the river.