Since you own the trailer now it will be your responsibility to replace the battery if it is dead.
You will need to pick up a multimeter and mearuse the battery terminals which without Shore Power turned on. The batteriery terminals should read 12.6-7VDC if fully charged. Then go turn on the shore power for the trailer and then the DC VOLTAGE on the battery terminal should now read 13.6VDC telling you the converter/charger on your trailer is working normally.
Chances are your battery has its fluids boiled out if it hasn't been checked in a long time. Your older trailer does not have smart-mode charging features and having 13.6VDC always on the battery terminals will boil out the fluids over time. sometimes as quick at a month or so. You need to pull the battery caps and check the fluid level in each battery cap.
since this trailer nevers leaves the camping site it may not even have a battery installed. That however doesn't answer why 1/2 the lights are ON sometimes. It could be the base which is providing your power is having brown outs during the heavy usage periods...
Two sources of DC can operate your ceiling lights - one being a connected battery and the other being your on-board converter/charger unit being connected to 120VAC. Your ceiling lights along with all of the other 12VDC appliances run from a 12VDC Power Distribution Panel in the trailer where all of the ATC type DC FUSES are located.
A multimeter reading your DC VOLTs level which is what runs the ceiling lights in all trailers will tell you the story.
Another useful thing to have with trailers is one of those RV120VAC VOLTMETERS.. You can have this plugged into one the trailer 120VAC receptacles and can tell at a glance what the 120VAC coming from the camp ground pedestal is reading..
Looks like this... SHould be able to pick one up from a RV place there in Montgomery or order one from AMAZON... I noticed a couple of RV places on the East BLVD... All is fine if the voltmeter is always reading inside the "GREEN" scale.
You should also pickup one of these multimeters from LOWES/WALMART very inexpensive $5 to $20. This is must have for RV'ers that have batteries.
Love your town of Montgomery but have noticed it has become somewhat of a WAR ZONE now on SOUTH BLVD between I65 and I85 intersections... Wouldn't want to be down on south blvd at night where all of the old but goody Hotels are that are now all boarded up...
My working day travels brought me down that way many of times from the mid 60s to present times working at a Space Surveillance station near Wetumpka...
Roy Ken