I don't see how you can fold here. But I also instinctively do not like shoving here either.
Let's work backwards though to see why...
After the turn
The 2c is an interesting card in that it is either a complete blank or it took you from being way ahead to being way behind (i.e., villain hit the flush). At this point, there is $145 in the pot, and villain has $150 left. Villain puts 1/3 of remaining stack into the middle, so it's $50 to you to win $195 (not quite 4 to 1).
Now, with the A
you have the draw to the nuts PLUS you already have what might be the best hand if villain doesn't have two pair or the baby flush already. We cannot immediately presume that villain has 2 pair+ here. That's playing "monsters in the closet" and it's a losing mindset unless you have a pretty good read on the guy from past play (I have no idea since there is no history in the lead-in).
By the turn, I just don't see how you can fold... nor can villain fold. If villain already has a flush, you have 7 outs, giving you a 15% shot to win - which isn't good - but that's only if he has exactly a flush already (and remember, you're getting 4-to-1 on a call here, too, so you're almost priced in). He may also be betting with a naked K
here. He has a wide range of plausible hands given the line he took. That said, I'd probably call here and re-evaluate the river.
On the flop
You have TPTK with a inside straight/broadway draw and backdoor nut flush draw. You were also the pre-flop raiser and the pot is $75 with you facing 4 villains from MP. You have to bet at least $50 here. I'd probably bet closer to the full pot in fact. By only betting $35, you allow anyone with a mediocre draw to stick around cheaply for implied odds and you aren't maximizing what you can win if a scare card hits the turn (which it does).
Pre-flop
Without much background, I have no idea if $15 is a standard raise here, but it seems fine. Getting 4 callers is a bit infuriating though. Do you have a LAG image or something?
---
In the end I call the $50 to peel the river and re-evaluate. I just don't see how you can check-fold here with a made hand and a good chance to improve.