house batterys not charging with engine running

Check voltage output of your alternator. Is it about 14 volts? If so check voltage to chassis battery solenoid are you getting 14 volts on both sides of large terminals or just one side?

Do you have 12V power when stopped and not plugged into shore power? In other words, are the house batteries actually connected?

Fairly common problem is that the solonoid that connects the coach and engine batteries when the engine is running, will both charge then when the ignition is off, seperates the coach battery from the engine battery so the coach will not discharge the engine battery. The manual switch brings them together to start if for some reason the engine battery is dead. I have had to change that battery isolaton solenoid on every camper I had (4 over the last 20 years or so) the only one that did not fail whas an old camper that used a solid state isolator instead of a solenoid.
I suggest you replace the solonid with one rated for continous duty availabale commonly at NAPA auto parts stores. Do not replace with the same trombetta solonid it probably has. It will not last.

Two pieces of test equipment I use that allows me to be in the cockpit and test voltages in the battery box or other places separate from the cockpit are a good clip on test light and a piezo speaker or siren. I use the piezo a lot. Clip one lead to ground and one to the actuation terminal of the solenoid and then go to cockpit and turn the key on and then start the engine. You'll hear the piezo clearly in the cockpit if the circuit is good.

Carry on from there.