I think you have two separate problems.
When driving the chassis batteries should be charged direct from the alternator. I would suspect your alternator is failing. A bad diode would give that sort of symptom.
When on shore power, there is a charge controller that monitors the state of charge and closes a relay as required to direct charging to the chassis batteries. The basic logic is that the chassis batteries must be below a voltage threshold and the house batteries above a certain threshold for this to happen. It could be the charge thresholds aren't being reached for some reason (unlikley with new batteries) or the charge controller is malfunctioning. On my '04, the controller is an Intellitec BCC device and a component failed on its circuit board, preventing charging of the chassis batteries. I think your '02 has a different type of controller, but the failure could be similar.
You could bypass the shore power charging problem by adding a Xantrex Echo Charge or a Trik-L-Charge. Either device will keep the chassis batteries charged while on shore power or generator.
Former owner of 2004 American Tradition and several other RVs
Home is West Palm Beach, FL