The other leak isn't really a leak, it is a siphoning action. Look into the tank. Just to the left of center you will see a pipe, about 1 inch in diameter, sticking up above the water line. It is called the overflow tube. There will be another tube, about 1/4 inch in diameter, sticking into this 1-inch tube. The little tube, called the fill tube, should only extend just into the top of the big tube. If it is pushed down too far, it will continually siphon water out of the tank into the bowl and on down the drain. With the help of a flashlight, you will see water dripping out of it and can, sometimes, even hear the dropping.

If this is your problem, just lift the small tube; often it will stay in place toward the top of the large tube. If not, a paper clip will work for a while. Place the small loop on the clip inside the small tube. Hook the free end of the clip over the large tube. Check this from time to time since the clip will rust quickly.. Replacement tubes, and ball cocks now come with a clip to hold this tube in place.