Some good and some not so good but still accurate responses. Kind of odd.
Batteries: Average battery life if properly maintained is 7-10 years, Mine made it 9 but.. Well they really should have been replaced sooner, yours may be 8, so that could be it
When driving as many have noted the engine alternator SHOULD be charging the house batteries,,, Unless, of course, you disconnected the house batteries (some folks do) in which case that may or may not have happened depending on who wired your RV. And in some cases I mean WHO, as in which technician at the factory.
And as someone mentioned, Many DP's have good size inverters, Heck my Gasser has 2,000 watts. If I do something like switch the switch that lets me use the inverter to power the fridge.. My Alternator can NOT keep up with the Fridge. (350 watts) and other loads.
Residential fridges are a bit less power hungry as a rule (With the fridge on Propane no tribble at all).
And of course Isolator systems can and do fail, there are two common and one uncommon types
Type 1 (Most common) is a solenoid, may be bi-directional or mono-directional (matters not for this discussion)
Type 2: Diode pack
Type 3 (rare) dual alternator one for each bank.
Home is where I park it!