Hello clandestineAl,
Preflop: Nah, it's not that wrong to Call vs BU who definitely will even have worse Qx/Kx hands in his range. Another option would be to 3bet depending on the opponent.
As played
Postflop: Well, depends really what kind of opponent he is, do you expect him to 2nd barrel here? If you do then Checking the turn is fine, another option would be to even donk the turn to protect your hand. But as played I'd definitely bet it smaller to give him room for hero-calls with weaker pairs, there ain't gonna be a lot worse hands which will pay you, I'd bet ~$0,80.
Best Regards.