J&M- look on the Vansco multiplexer module on the front right side of batt compartment. Is #19 LED lit on the right hand bank of lights? If it is, the charging solenoid on the batt compartment back wall is probably bad and not connecting the chassis batt's for charging as it should. If no #19, there is a problem reaching full charge on the house batt's (#19 & the solenoid connection to charge chassis batt's require about 80% charge on the house batt's before energizing the solenoid to charge the chassis'). The short answer to your ?? is yes, shore power is supposed to charge both batt banks if all is working.
Here is an illuminating discussion of electrical problems
with some good insights on charging, and one on Chassis Batteries
w/similar issues and a link to process for testing the charging solenoid.