Somewhere you have a readout of your house battery voltage, often by the battery disconnect or your inverter/charger display.  When disconnected from shore power these should read battery voltages somewhere around 12.5 to 12.8 volts or so.   Now start your engine and soon, it might be a few minutes, the voltage will jump into the 13 plus range if the alternator is charging the batteries. A multimeter on the battery leads set to read DC voltage also works if you do not have a voltage display.  When on shore power your inverter/charger should be keeping the battery voltages around 13.2 to 13.5.
It might be a loose connection instead of a solenoid if the alternator is not charging the batteries.

As an aside a rule of thumb with alternators is to not run them long at over half load, so a 165 amp alternator should not run more than a few minutes providing more than 80 to 85 amps of charging.  A TV or refridgerator on electricity (some inverters are wired this way or change it to not use propane while driving down the road) is way under these numbers, but don't run your microwave/convection oven while going down the road expecting your alternator to handle the inverter load without burning out your voltage regulator.