Even though the rubber flapper looks and appears seating properly be sure to use a rag and wipe clean all the scum and foreign matters that the rubber flapper seats on.

Did you use a brand name flapper for replacement or an unknown brand?

The coarser back of a dish washing sponge works great for burnishing off the unseen mung on flapper valve seats. A light touch does the trick.

The identical replacement at my local hardware store was Ace Hardware brand. It was the only one exactly like my old one.
I tried cleaning the surface where the flapper rests with a cloth. I'll try something coarser.

The Fluidmaster reds and the Korkys seem to be very reliable.

I cleaned the surface, and I think the water is getting by somewhere else. But where?

I have seen instances where a new flapper just didn't seal. Try a new flapper.

Is the water in the tank rising to the level of the overflow tube?

There is a possibility of a hairline crack in the seat you cleaned with the rag.

First guess is the flapper and seal.

I have seen situations where the valve was replaced and the fill tube was too long. Instead of trimming it to length, the previous person just pushed it down the overflow tube. The result is that it slowly siphoned the water from the tank.

Since this is a new valve, check to see if the fill tube is too long.

Thank you, all! I returned the flapper that would not seat properly, and replaced it with a Fluidmaster flush valve Repair Kit. Seems to be a way better system. My desire to stay with as close to original as possible, wasted a lot of time and caused a lot of frustration.

Usually the most obvious solution is the correct one.

It'a hard to beat the Fluidmaster stuff for price and reliability.