1. The stock battery monitor will not show true battery condition when connected to shore power. You must disconnect shore power for several hours and then check the battery. Even then, the stock battery monitor is, at best, rudimentary. Measuring the voltage with an inexpensive digital voltmeter and comparing it to a charge status chart for the type of battery you have is better. An actual charge discharge monitor is better.
2. I don't know specifically about your trailer, but on my 2001, the disconnect switch leaves the propane detector connected. The newer ones only draw about .04 Amps, or about an Amp-hour per day. If you have two group 24 batteries you should have about 170 Amp-hours of total capacity, but should only plan on using 50% of that to avoid battery damage. So good batteries running only a propane detector should still work after 6 weeks to 2 months. Since you didn't turn off the disconnect switch, I can't say how long it should have gone - another member will have some ideas. Refrigerator, furnace, stereo, TV, are all possible parasitic drains. Storage compartment lights are famous for getting bumped on during travels. The bulbs in my trailer drew 2 amps. Only one such light left on would drain your batteries in less than a week.
3. It is not uncommon for trailers which have been left on shore power for extended periods using the stock converter-charger to have damaged batteries.
4. The best test is to hook up to shore power for several hours or overnight. Then disconnect the shore power and periodically test the battery condition with the monitor panel or a voltmeter.
5. Your jack should have worked once shore power was connected. If it didn't you may have an intermittent wiring problem. Mine worked fine for several months and then quit. The negative cable from the jack pulled out of its connection on the battery negative bussbar.
"You cannot reason someone out of a position they have not been reasoned into"
Al, K5TAN and Missy, N4RGO
2002 Classic 30 Slideout
2001 Safari 25 RB Twin (Gone, but not forgotten)
2013 Dodge 2500 Laramie 4x4 Megacab Cummins
WBCCI 1322, TAC FL-39, AIR 82265