As Drew said, the water leaking back out of the city water inlet means a bad check valve in the inlet assembly.
Some RVs have only a gravity fill line to the fresh tank (sounds like that is what yours has, and it's kinked), others have a valve that directs the city water hookup into the tank (city tank fill), and some RVs even have both. In the case of a "city tank fill" valve, they can sometimes go bad, leaking water from the city hookup into the fresh tank even if the valve is turned off.
The only other way for the fresh tank to fill while using city water is a leaking check valve in the water pump itself. That sounds like your issue.
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