How to best play a flushdraw or OESD?

You played it fine, could also check back on the flop. Wondering what he decided to minraise on the flop and check two later streets. After the check on the river I would definitely put in a small value bet, maybe around 8bb, its unlikely he decides to minraise backdoor clubs and the only hands we are beat by are 23 or 67.

Nut FD, easy 3bet for me, shove on the turn.

"I'm a bit unsure how to play a flushdraw or when I have an OESD. "

Nut FD and non-nut FD are very different hands. Also the OESD. You must look for dominated draws and fold equity. Also if you choose to call, you must consider for the reverse implied odds. And the regular implied odds of course :). Please, give another example.

Poker Stars, $1 Buy-in (60/120 blinds, 15 ante) No Limit Hold'em Tournament, 9 Players
Poker Tools Powered By Holdem Manager - The Ultimate Poker Software Suite.

SB: 2,865 (23.9 bb)
Hero (BB): 3,285 (27.4 bb)
UTG+1: 6,294 (52.5 bb)
UTG+2: 10,137 (84.5 bb)
MP1: 2,904 (24.2 bb)
MP2: 9,931 (82.8 bb)
MP3: 6,173 (51.4 bb)
CO: 4,634 (38.6 bb)
BTN: 4,182 (34.9 bb)

Preflop: Hero is BB with J♣: Q♣:
2 folds, MP1 raises to 240, MP2 folds, MP3 calls 240, 2 folds, SB calls 180, Hero calls 120

Flop: (1,095) A♦: K♣: 5♣: (4 players)
SB checks, Hero bets 679, MP1 folds, MP3 raises to 5,918 and is all-in, SB folds, Hero calls 2,351 and is all-in

Turn: (7,155) 4♣: (2 players, 2 are all-in)
River: (7,155) 9♣: (2 players, 2 are all-in)

Results:

Not a bad play. Big donk bet with FE and then call with enough equity. But there is two problems on this board. First, we are against Ax heavy range. Even Kx is here. Yeah, we block many of this hands. Less FE, but I think still enough to be profitable. Second, what if he calls on the flop? What we do then? Shove on the turn. Or check-call. Seems ugly.
I prefer check and then call or check-shove. Depending on the size. Also we cant raise vs two players. You can call even if somebody shove directly, because he will not do that with nut FD, most likely. Sometime we must fold on the turn. If he bet small and then shove. Shit happens.

The min raise on the flop by hand 1 is suspicious for sure, but I don’t think you have much fold equity with 3betting the flop. I would most likely call as well for pot control. The river is a value bet for sure. If he had a better hand he would bet the turn or river for sure.

Hand 2 is an easy call with the draw we have on his shove

Cheers,
SDK1987

1st hand you had 15 outs [9 hearts, 3 aces and 3 seven] and in flop you had enough pot odds. I understand that you can bet on turn e river to extract money.