The problem with the old toilet sounds like just the old rubber flapper was leaking. They may look "normal", but they all get weak and eventually start to leak from the tank into the bowl, and just need to be replaced periodically.
You didn't necessarily need to replace the toilet.
Take the tank lid off and make sure that the refill assembly is properly adjusted so that it cuts off at the horizontal water level indentation inside the tank, or about 1/2" or so below the top of the overflow pipe in the middle of the tank.
If you have a float-cup type, you can adjust the water level by twisting the assembly shaft or pinching the float-cup clip on the little rod beside it to slide it up or down to adjust the water cut-off level. If you have a float-ball type, carefully bend the ball rod down to adjust the cut-off.
Make sure that the refull tube water flows only into the overflow pipe.
Adjust the flapper chain so that it just allows the flapper to close fully.
Toilets have built-in traps, and any time that the water in the bowl rises high enough, the bowl water will "leak" out of the trap into the drain pipes.
I think that you just need to do a little adjusting inside the tank.