Yep, there is a one way valve somewhere that is sticking and/or your water pump diaphram has failed.
In years past a anti-backflow valve was usually on the output side of the water pump to prevent that from happening. Later on builders eliminated that valve and depended on the water pump head to stop the flow.
FWIW: I can't say that I know what a Skyline Aljo 19LT is, or how old it is, but it doesn't sound like any of the Molded
Fiberglass trailers that this site usually sees.
A new valve or pump head should be avalilable at most RV parts sources
But I am certain that the same answer about the water system will apply.