Pump Runs Continuously

I tested our pump and it does not seem to prime correctly. When I fill the freshwater tank - then turn on the pump it runs and does not build pressure. As a test, I turned off pump, connected TM to city water and built pressure, then turned off the city water and turned on the pump again. The pump stopped running like it should. This tells me I don't have a leak. If I open the faucet the water runs briefly then the pump starts up again with no pressure. I noticed a few drops of water under the pump. I am going to check the screen between the tank and pump, then take the pump apart. Any advice for me doing this? Thanks a million for your ideas!

It takes me about 5 minutes to purge the air out of the lines when using the pump.

After filling the water tank, the water heater still has air in it. It will take six gallons of water to displace that air.

I place a plastic container in the kitchen sink and open the hot water up. This will blow a lot of air and perhaps a few drips of water. When you get a steady stream of water it means that the air has been purged.

Alternatively, I sometimes open the two drain valves at the low point on the rear of the TM and let the air out there.

At any rate, eventually you will need to let the air out of both hot and cold water faucets at the following locations:
kitchen sink
bathroom sink
indoor shower
outdoor shower
rear drains

If you leave any air in the system it will be very confusing to the pump.