"... the float activates and tank fills up an inch." Do you mean that when the water level drops (due to the water leaking out the flapper), the water only fills to an inch???
"The water level in the bowl seems to go down slightly too." Unless I'm missing something, this will happen 1) as water leaks out the flapper, and 2) when the tank flushes. You may need to say more about this.
Anyway, perhaps the flapper is not seated properly on the hole. It may be out of alignemnt either left-to-right, or not setting COMPLETELY down on the hole. It's centered properly but it's kinda 'tilted', or sitting 'high' towards one side. (What happens when you put slight finger pressure at various points around the edge of the flapper? At some point, does slight pressure stop the dripping? If so, you have your answer if it's tilted / high.)
Perhaps the rim on the flapper hole needs to be cleaned. If there's any debris/rust/gunk hanging up the flapper, it needs to be removed. Run your finger around the inner, top, and outer part of the hole to see if you feel any 'high spots'. If so, you may be able to pick them off with your finger nail.
Other than that, I don't know if there are different sizes for flappers these days. (Plumbing is not my forte.) Do find out. The make/model should be on the inside of the tank and/or lid. My best to ya and hope this helps.