Inside my toilet tank, the overflow pipe, which is a very thin plastic tube, broke off at the base. The toilet was "running", I replaced the flapper and it continued to run. Whole working in that tank, I noticed that this overflow pipe was crooked and when I touched it, it moved.
Immediately, I checked it, and found a crack at the base of it. A little hand pressure and it cracked right off. That crack is where the water was leaking, NOT at the flapper.
While replacing the whole flapper drain assembly is not a major repair, it does mean the tank has to be removed from the toilet base, and most of the time the bolts are corroded and can not just be unscrewed. Too much pressure applied to them, and the tank will break. So, that means it's usually a matter to saw off those bolts with a sawsall metal cutting blade, or by hand with a hacksaw blade. (Either way, it's a pain in the ass job).
I measured the hole where that plastic pipe goes, and went out to my garage to see if I had some sort of pipe to glue in there. I found that a piece of galvanized steel pipe was just slightly larger. I grabbed a piece and was able to slowly turn it and cut threads with the threaded end of that pipe, right into the hole. Once I had threaded it into the plastic about 5/16", I unscrewed it, coated the threads with silicone caulk, and screwed it back in. I left it overnight (without water) so the silicone could dry.
It works perfectly. The steel pipe is exactly the same height as the original plastic piece and it took me less than a half hour to fix. Since I had the pipe, it cost nothing.
So, if you even run into this same situation, you know what to do now...
This steel pipe will probably outlast the toilet. That thin plastic pipe which was there, had a wall thickness about as thick as a business card. NOT MUCH OF A PIPE!