If the valve itself is leaking, water would be rising (maybe quite slowly) until it goes down the overflow tube. If it never makes it that far, it's because water is leaking into the bowl, and that could be the flapper valve or its seat (there could be some buildup on the seat and it can't seal. Rarely does the gasket that holds the flapper assembly in place fail, but the nut (usually only on a new one) might be a little loose.
So, after sitting for awhile, and you hear the drip, look at the water level. If it is high, the valve itself, not just the float, needs replacement. Depending on the design, it may take a more modern valve and a Korky Quietfill, or a Fluidmaster valve would work. If that doesn't happen, shut the water supply off and leave it for a few hours and see if the level drops. If it does, then it's the flapper valve or its seat. Note, if your flapper came with a ring AND tabs, you only can use one OR the other, not both...if you tube has tabs, cut the ring out. Using both can distort the flapper, and it won't seal. Also, depending on how it was stored, some flapper valves may have become distorted, and won't seal, either. It's also possible, you just got the wrong one, or the chain is too tight or loose (loose, it can get caught underneath).