Shallow well pump

We have a shallow well pump connected to a pipe that was T'd so that we could turn valves to run water either from a rainwater cistern or a dug well. The pump (1/3 HP) went out about 10 years ago. Since we had switched to city water, we let it go. Recently we decided to replace the pump so we could run the water for garden/yard purposes. (there is an outdoor faucet that is the only plumbing running from the pump now) After installing the new pump (3/4HP with 7 gallon air tank and a 20/50psi switch) according to manufacturers directions, we tried to prime it. The pressure gauge only goes up to 30-35 so the pump never shuts off. If we turn the faucet on, pressure goes down to 0 and no water comes out. We have tried priming both with the cistern valve and the dug well valve separately and both act the same way. I have double checked the seal on the pipe that is visible and even have put silicone around any I thought might be questionable. The pump is in our basement, about 7-8 ft. below ground surface. The plumbing going into both the cistern and the dug well are below the surface of the water as when we remove the covers, no plumbing is visible. I don't know the age of the plumbing as we have only had the house for 25 years, but the house itself is over a hundred years old. My thoughts are that the problem is probably between the T and the pump, as both lines act the same, but then I'm no expert. Any thoughts or recommendations you would have would be greatly appreciated...Thanks

Of course you could have a bad pump. What brand is it and where did you buy it?

I've never heard of a 20/50 switch.

Maybe you have an old gate valve that isn't working right.

Are you priming into the pump or into just the plumbing?

How much water do you get when the pump won't shut off?

Pacific Hydrostar item #02955 pump, Bought at Harbor Freight. Stats- 115v, 5 gal tank, 898gph, 20-50 PSI switch, Max suction lift 26 ft., Max lift 140ft. I was attempting to prime from the cistern and/or dug well into the pump. The first attempt at the cistern we got about a gallon of water out of the faucet, but then lost pressure and flow. Haven't been able to get any water since. Opening the faucet just drops the pressure to 0. If it is a bad gate valve, then it is two bad valves as they both give the same symptoms whether I have the cistern valve or the dug well valve open. Just read on Harbor Freight's website that the switch is a 20/60psi, the box says 20/50.

Sounds like a suction leak. Probably coming through the handle of a ball valve on the suction. Things that will hold water under pressure sometimes can't hold out air under vacuum.