On my Triton I have two pumps one to fill and recirculate or pump out but all it does is fill the live well to the overflow and keeps pumping fresh water in. The other is the aerator that circulates what water that's in the live well stirring it up and it also has an air vent. Right now my aerator pump is not working so I don't know if it's the timer or the pump itself the light on the switch comes on but pump doesn't. When I was under the dash changing out my bad gages I didn't have my meter with me at the time so I didn't get to check it then. I rarely ever use the live well except in a tournament or the few times we go crappie fishing and are keeping them. I've been just pumping water out and pumping fresh water in and try not to let the live well get over 3/4 full for weight purposes. It's a big single well with divider.
Florite has rebuild kits for the valves. That's where I would start. Yours is probably like mine. The valve shuts off the fresh water intake and just recirculates the water. It may not be completely sealing.