Ok! More info helps cut it down. The check valve is not likely to keep the city water from the hose from entering, just works a s seal to keep water and air frompassing out as the pump is used. Tiny little spring presses it out against the metal and water pressure will move it back to enter.
When on city water, do you feel or hear any water moving in the hose or simply like it is turned off and no movement?
The drawings for your RV are not the best interactive type for after 2010 but take more study.
Drawing here for drain and fresh water:
https://www.winnebago.com/Files/File...36ld_plumb.pdf
This is a snip of the fresh water in going to the diverter valve for change from fill to normal use as I've marked in blue. Check this before trusting???
Click this drawing or go direct to get a better view....
If the valve is working right water comes in and goes pretty straight to the cold of the shower as well as tee for other parts of the RV. I marked that in green?
If I have the right valve, this can happen and might be the only thing I can think of to stop water that goes so direct.
This valve is often plastic and the shaft can break as we turn it, so that the inside is turned some but not fully in either fill or use position! Water can't go either direction, so we can get what it sounds like you find???
Feel free to correct if I'm missing reading it right!!