Start by removing grounds from any cables in the system, cleaning their connection points on the metal frame, inspecting the ground connector & replacing the cables where warranted. Clean and water the battery cells. Buy a digital voltmeter if you don't have one already ($8).
With all the grounds being good, than measure the battery voltages at various points in the system and list them here, starting across the batteries. If any battery is under 12 volt, charge it up before continuing with the tests.
Batteries charged up, disconnect from shore power, start engine, measure chassis battery voltage. Wait a couple minutes for the timer to engage the crossover solenoid and then measure the voltage across the house batteries. Record the voltages. They should be above 13 volts. Have a rubber mallet nearby to whack solenoids that need it.
Stop the engine, and plug the RV into shore power. Repeat the voltage tests. You should have 13+ volts across both sets, the house and chassis sets. If you have it on the house but not the chassis batts, there's your problem. There isn't an auxiliary charger for your chassis batt, which is typical for 10's of thousands of RVs. $20 at Walmart.
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As far as your genset not starting, check your dash for a AUX or Emergency Start Switch. Push that while trying to start the genset. If it starts, you just have a discharged battery.