Top two pair v. possible made flush/straight

I figure should have PFR .50. Would probably still PSB the flop and then tried to get thru the T as cheaply as possible given top Two Pair against a poss made flush/straight. When I didn't boat on the R, should I have folded to the push?

Ultimate Bet
No Limit Holdem Ring game
Blinds: $0.05./$0.10.
10 players
Converter

Stack sizes:
UTG: $4.27
UTG+1: $2.41
UTG+2: $4.80
MP1: $11.67
MP2: $9.35
MP3: $13.83
CO: $14.97
Button: $9.25
SB: $3.57
Hero: $5.86

Pre-flop: (10 players) Hero is BB with Q :heart: A :heart:
UTG calls, 2 folds, MP1 calls, 4 folds, SB calls, Hero checks.

Flop: Q :club: T :club: 3 :club: ($0.4, 4 players)
SB checks, Hero bets $0.4, UTG folds, MP1 calls, SB folds.

Turn: A :spade: ($1.2, 2 players)
Hero checks, MP1 bets $1, Hero calls.

River: 6 :spade: ($3.2, 2 players)
Hero checks, MP1 bets $4.35, Hero calls.

Results:
Final pot: $11.9

Nothing wrong with checking AQs in the big blind sometimes = trap.
Flop, yep, standard line and good.
and then the bad part...
Turn. WHY are you checking here? If villain is drawing then you just gave up a free card. You're making it look as though you are scared of the ace and now you have no idea if villain has AT or A6 (my read), JK, or the flush. Bet close to pot here. As played, you're in a very difficult position on the river - I would fold, angry with how I'd played it...

hmmmm.

I quite like it and sorta dont. hmmm.
instinct says bet turn and river outright.

Had another one last night similar to this. Flopped top 2 w/ AQ on a board that had 3 suited by the turn. Had PFR'd, was now HU OOP, c-bet PS on the flop and was called, and the pot was now around $4.80. I had $2.35 behind me. Do I 1) bet into the Turn, which really has to be a push to make it remotely close to meaningful, knowing I'm just handing him chips if he has the flush as I only have 4-outer on the river, or 2) check and fold to a large bet, altho' with this I figure I might as well just put up a big sign saying "I'm scared of the flush". Don't like either option, but I keep running into this situation and not real clear on the best line.