It doesn't seem like effective stacks are deep enough for this kind of play. If the pot was headsup to the flop at 2,600 that must mean blinds were about 200/400. (400x2.2 = 880 x 2 = 1,760+400+200=2,360 plus antes=2,600). and if he had 2,000 left after calling your 2,000 flop bet and calling your 880 preflop bet then he only had about 5,000 to start which is only about 12BB.
Preflop: raising and folding are both OK, again without knowing your stack size it's tough to say. But if you were also in the neighborhood of 12-20bb then folding is better. If you're raising a marginal hand from EP you have to believe that you can usually outplay your opponents post flop OR the blinds are tight and the table will give you credit for a real hand fairly often to make this profitable as a steal. Stealing from EP is pretty tough, so generally you will be playing this hand for it's post-flop playability....and for that fairly deep stacks are generally required.
Flop: that is a pretty decent flop for our hand. But, it is also a pretty good flop for our opponent's range. assuming he's got 4,000 behind if you Cbet pretty much any amount and he jams, you'll be priced in to call with your 12 outs twice. (If you Cbet half the pot for 1,300 and he jams 4,000 into 3,900 you'll only have to call 2,700 to win 7,900 so you're getting 3:1. Pretty much mandatory call). So I don't like Cbetting. If you're forced to call a jam anyways, I'd rather take that move away from him by jamming myself. But jamming doesn't make much sense either. risking 4,000 to win the 2,600 in the pot and you'll generally only get called by sets, 2pair and AK or better combo draws. so shoving the flop isn't a great choice either. Which leaves us with checking. he might check behind giving us a free card at our draw. Or he might bet, opening us up for a check raise jam and realizing the full
equity of our hand PLUS some fold equity. If you put him on a Jack, he would probably fold a jack to your check raise. He would probably also fold weak aces....AT would be borderline.
If you check the flop and he checks behind playing the turn will be just as tricky as the flop; except now you've lost the betting lead and your draw is a LOT weaker. So, I'd be inclined to check/fold the turn. Weak line, I know.
Because of all of these problems, maybe you are beginning to see why raising with marginal
hands in early position and shallow stacks is generally not recommended. Good options are hard to find, even on pretty good flops. But out of all of those, check raising the flop seems to be the best option.