On the backside of water heater top line (hot out).....disconnect hot line.
Inside the fitting that is screwed into tank is a check valve.
It is plugged.clogged/jammed shut from all the calcium/stuff from AZ Hard water
You probably will need to replace it.
You also need to drain/flush water heater more frequently.....especially when using hard water (AZ, NM, TX, S/E CA, etc.)
Anode rod doing it's job...protecting WH Tank. But in doing so is dissolves plus the crud from the hard water.
And YES......city water and pump discharge are tied together. So both feed everything cold/hot.
Basic RV plumbing system.