" Water will run out of the pressure relief valve whenever it cycles on"
Are you referring to the valve near the outside water inlet connection? Or, ??? whatever valve you're referring to, if water is being emitted, there is a problem in that area, it would seem...
I replaced the original water pump in our 78 Ambassador with a current model variable flow model that is far superior to the old unit.
This new pump matches the flow required by the valve being opened - cracking a faucet a small amount results in a quiet, low speed from the pump - opening both the bath & kitchen faucets causes to pump to speed up to match the need - works great!
BTW - I had very low flow from the HOT side of the kitchen faucet when we bought our AS a few years back... I cut the copper tubing near the faucet on the HOT side and found some small 'balls' of hard crud (probably from water heater that wasn't flushed annually) lodged inside... I took the faucet cartridge and tried to 'blow' the crud out with an air hose without success... Ended up replacing the entire faucet with a new Moen similar unit - Ahhhh, mucho hot water for the dishes now!
Ray & Pat; Morada, CA