How to adjust water pump pressure, pump cut-on pressure and pump cut-off pressure - private pump and well system do-it-yourself repairs
Switches can be purchased separately and are cheap. The site above shows how to adjust one. I have mine on 40# and 58#. I would adjust the switch before trying anything else. Remember to turn off the breaker while you're turning the nuts on the switch. The nuts themselves are not "hot", but it's easy to lose focus and brush a wrench against the contact points. I did that one time with the power on and don't care to do it again.
It might be that your pump is having trouble getting the pressure up to the cut-off point. If that's the case, then a new pump would probably help. It does not sound like your tank is the problem since you can run water for six minutes before the pump has to cut back on. Generally speaking, you should be able to get at least five or six gallons out of your pressure tank before the pump cycles back on.
By the way, when you are using water, can your pump supply enough water to keep the faucet running and still pump up to the cutoff point of 52#? If so, then I would have a hard time believing the pump is the problem.
Since you refer to rusty fittings, I am assuming that your pipe is galvanized. Just be aware that galvanized pipe tends to develop mineral and rust deposits on the inside and that can impede your flow.