once turning 3rd/3rd with backup, I think you're calling down.
bet/calling flop can't be correct, though...
those bare non nut wraps in both directions are good candidates for checking behind on the flop.
once raised, you probably have around 35% pot equity, not quite enough given effective odds and having to fold to a bet after many turn cards, depriving your opportunity to see the river, which a good chunk of your pot equity lies upon.
when getting raised makes it close and leans on the side of folding out non trivial pot equity, you often do better by checking behind.
we know from Hwang's PLO books that even a small pair that accompanies your wrap adds a lot of value to your hand. indeed, making the K a 2, your equity jumps to 40%, in which case, I like your line a lot better.
not trying to state the obvious: "I like better hands better!", rather, that I like to visualize my range and decide which type hands i'm going to c-bet and which i'd normally check behind, assuming an unknown villian, whom I can't exploit quite well enough, yet, forcing me to revert to some type of GTO approach.
I also don't think checking behind makes life easier for villian.
you are going to call a bet on most turn cards, using position and the fact that now you're playing a controlled pot against a much weaker range, allowing you to comfortably value bet a river K, for example...
trying not to be results oriented here: "oh, you got check/raised, let's check, then!". not that! also not saying you played this hand badly or anything...
just something to think about
: c-bet often, but not all the time, and don't just randomize your bets and your checks. use common sense as to which hands are worth checking (usually those of marginal value and decent, but not great, pot equity). food for thought, I suppose.