If water is "constantly dripping" out of an overflow that is connected to the F+E tank then in my experience there is a good chance the ball valve has failed because if it were working it would have shut-off tight be with the level safely below the overflow port.
It is normal for the water in the F+E to be warm but if it's really hot then presumably the system has been pumping over. The hot water in the F+E tank could be the root cause of the float valve failing or they could be independent of one another.
Or maybe the overflow is not connected to the F+E tank and the problem lies elsewhere.
If you're not confident and trained for this sort of work, get an experienced plumber involved. Mistakes can be costly and, you never know, if they like the standard of your normal work they might repay the favour and sub some work your way in return.