The easiest way to tell is use your multimeter set to DC VOLTS and measure across the battery terminals with the battery hooked up normal. If the battery is fully charged it will read 12.6-7VDC.
Then go plug-in the shore power cable to 120VAC. Then read the battery terminals again using your multimeter and it should read 13.6VDC telling you the converter/charger in your trailer is charging your battery..
Your inexpensive multimeter ($7-$20 from WALMART/LOWES) will look similar to this with two probe leads...
Rotate the center knob from OFF to the DC area to a voltage setting of 20VDC or more.
Touch the probe on the two battery terminals and it will read the DC battery voltage on the scale..
Will look something like this on the digital readout...
This will be reading 12.6VDC if you have the DC VOLTS selected on the lever switch. If your battery reading is below 10.5VDC then the battery is at zero percent charge state. When most batteries get below 12.0VDC and are not immediately re-charged back to their 90% charge state (12.5-6VDC) they will go bad on you pretty quick.
Be sure to go back to OFF on your multimeter when done reading as some multimeter do not have auto shut down and will drain the internal battery if you don't turn it off when not in use.
If you do not see the increase of DC VOLTAGE goto 13.6VDC then there is a disconnect between the Power Distribution Panel in the trailer and the battery terminals. This can be a blown fuse near the battery or two blown fuses on the power distribution panel labeled REV POLARITY or a bad connection at the battery terminal.
Learning to use your multimeter will pay you back many times over. It takes the guessing out of things...
Let us know how you make out - will walk you thru the things to do to test it again...
Roy Ken