Well you have done a lot of good trouble shooting already. Initially I thought bad pressure valve in the pump, but then you drew water from the external bottle via winterization line with a capped output on the pump.....and the pump stopped running and held pressure. I assume your pump filter has been cleaned and is located between the pump and your winterization line / freshwater line T, which is typical......which would prove its not leaking air. So to me it’s pointing back toward the fresh water tank either not full enough with water or air leak in the feed line before the winterization valve (which is what you suggest also I think). Make sure the fresh water tank air vent is not blocked with water or just open the fill cap for your pump testing (so no vacuum forms behind any water in the tank). The fact that water is flowing out the pump in one of your tests may be mostly gravity feed perhaps? On my coach there is a shut off valve on the fresh water tank (gravity fill FWT also) and a shut off valve on the winterization line........so I have to close one and open the other depending on which liquid source I want to use.
To check the backflow valve in the pump you should connect the pump to the normal coach lines; turn on city water and open the winterization line. Make sure the FWT line is closed. any city water coming through the winterization line means a backflow valve problem.
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