Tom,
Your plumbing system is the same regardless of whether you are on city water or pump (onboard water). The only difference is where the water pressure enters the system. The outside shower that is referred to above is the fold out facet located in your plumbing bay. A 3 GPM water pump is minimal for your coach, but if it is operating properly, you will have reasonable water flow at one facet.
From your description, it sounds like you have a loose connection, probably at one of the valves in the plumbing bay or in the water pump compartment. If the auto fill valve was leaking, your fresh water tank would overfill when you were hocked to city water. You need to fix the water leak ASAP, because it will cause damage to the wooden flooring in your basement if it continues. If your pump strainer is clean and your pump is not running when the facets are turned off, you have a bad pump.
If you need to talk through you problem, just call me (713-254-4156). I am in the Mountain Time Zone now.
Gerald