Check the following:
(1) the chain length -- and make sure that the chain isn't pulled a little when you put the lid back on -- the chain should have a little bit of slack, but not so much that it bunches under the flapper. If it is too short, it can hold the flapper a little off the seat of the flush valve. Sometimes happens when the trip lever handle rotates a bit in its hole. Does the problem occur when the chain isn't hooked to the trip lever arm?
(2) whether the refill tube is just shoved down the overflow riser; if so, it can cause a siphon. It should have daylight between it and the top of the refill tube, but putting it on the little clip.
(3) whether this is the right-sized flapper. Generally, for an AS toilet with a standard flush valve, a classic Korky flapper should work fine.
(4) whether you have any cracks in the overflow riser, either above or below the flush valve seat. Turn off the water to the toilet and see how far it falls on its own. If it stops, say, 1/3 down from where it started, then check for a crack at that height. If it drains all the way out dry to the very bottom of the tank, below the flush valve seat, crack below flush hole. If it drains right to the flapper, probably a flapper issue as you suggest. There are seat repair kits, Korky makes one that comes with a new flapper and a little ring for it to seal against, but this may not be your problem, so try the other stuff first.
Good luck and let us know if none of this works.