I would remove the drain plug and put some teflon plumbers tape around it. That is, if you're unsure where it's leaking from. If you're sure it's not from the drain plug, you might have to remove the tank to find the leak.
If it were mine, and there was a tank leak somewhere relatively flat on the tank surface, I'd just use a piece of Eternabond tape to seal it.
The pump running like that does indicate a leak, and the water tank is the main suspect, but if you're sure it's not the tank, check behind the toilet for a leaking valve.
