[5NL] Playing TPGK OOP

This was villains 3rd hand so nothing to go on. Not too much interested in villain, more interested in how people would play TPGK OOP in this situation.

PokerStars - $0.05 NL (6 max) - Holdem - 6 players
Hand converted by PokerTracker 4

BTN: $3.87
SB: $5.26
Hero (BB): $5.11
UTG: $5.00
MP: $5.08
CO: $3.76

SB posts SB $0.02, Hero posts BB $0.05

Pre Flop: (pot: $0.07) Hero has K Q

fold, fold, fold, BTN calls $0.05, fold, Hero raises to $0.15, BTN calls $0.10

Flop: ($0.32, 2 players) 2 3 Q
Hero bets $0.20, BTN calls $0.20

Turn: ($0.72, 2 players) 8
Hero bets $0.40, BTN calls $0.40

River: ($1.52, 2 players) 6
Hero bets $0.80, BTN raises to $2.00, Hero ???

This is a typical spot I find myself in and I think it's one of my biggest leaks. How would people play this hand?

I don't like leading out on 3 streets with only TP but I don't like the idea of c/c or c/f either. I feel I can at least do my best to control the pot if I do the betting and if I get raised I can get away from the hand. Thoughts?

---------------------------------------------------------------------------------------------

Gonna post this one up as well as it's similar but obviously this time I am IP.

PokerStars - $0.05 NL (6 max) - Holdem - 6 players
Hand converted by PokerTracker 4

UTG: $5.00
MP: $5.44
Hero (CO): $6.08
BTN: $11.97
SB: $5.00
BB: $5.62

SB posts SB $0.02, BB posts BB $0.05

Pre Flop: (pot: $0.07) Hero has K Q

fold, MP raises to $0.15, Hero calls $0.15, fold, fold, BB calls $0.10

Flop: ($0.47, 3 players) 7 8 K
BB checks, MP bets $0.55, Hero calls $0.55, fold

Turn: ($1.57, 2 players) 5
MP bets $1.50, Hero ???

Thoughts?