A couple of things to check:
There should be no tension in the chain from the flush lever to the flapper when in the closed position.
also,
The water level must be a bit below (abt 1/2")the top of the overflow tube when the tank is filled. There's an adjustment on the float connection for this purpose.

I have adjusted the float connection like you suggested. The leak is flowing less, but still there. Should I continue to adjust lower?

Not all ballcocks are non-syphon. Raise the tube in overflow above water level and see if that helps.

Also, when it comes to brands, not all universal flappers fit well with all brands (like Kohler). The seats are slightly different. You would need to get a flapper for that particular brand.

If this is an older toilet you may need to remove the tank and change the flush valve assembly as they can become damaged over time.

As posted, there must be slack in the chain with the flapper seated and the fill tube thay goes in the overflow tube must terminate higher than the tube and the water level must be below that overflow tube. The correct water level is usually marked inside the back of the tank with a horizontal line.

If this is an old style water waster toilet then a red Fluidmaster (non-adjustable) flapper valve should work.

If the toilet is a newer 1.6 flush then you may need the exact OEM flapper regardless of what the aproned know-it-all at the big box store tells you or what it says on the package.

Shut off the water to the toilet and flush. Hold the flapper up or remove and take the scrubbie side of a wet kitchen sponge and softly run around the flapper valve seat in a circular motion. It may seem clean but harness deposits are hard to spot on a valve seat. If there are hardness deposits white vinegar will remove them.

Thanks for your suggestions. This is what I have done so far on my 1.6 American Standard: Flapper chain tension-ok, Overflow tube-above water level, Flush Valve-newly installed, Flapper-new and cleaned flapper seat. Also, lowered the water level. I think this leak has to be the flapper (Korky "universal"). today, I plan to purchased another flapper and re-clean the flapper seat and see what happens.

Get a real American Standard flapper valve.

Just wanted to follow up. The new flapper did the trick. I don't know what the problem was with the Korky... I purchased a FluidMaster universal flapper (the only other option at the Home Depot) and it worked.Awesome. Thanks for all your help, suggestions and encouragement.

Keep your receipt. If the water bill is outrageous because of the leak, many cities will modify it downward if proof that the leak has been fixed is presented.