OK ... here's what I've been able to check today, and the results:
CONNECTIONS: The two batteries (which are actually "valve-regulated lead acid", not "gel") are connected properly.
CURRENT CHECKS: With the two batteries disconnected, the DC volt reading is 6.3 on each, so that's OK. With the batteries wired back together, the reading is 12.6, so I think that's OK. With the trailer plugged back into shore power, the output current to the wires at the battery (tho' not yet connected to battery) is about 12 1/2 volts, so that seems normal.
So, maybe my display panel showing battery status is bad. One other thing that has always been the case, and may or may not be related to all of this is this: When I plug the trailer into a GFI protected outlet it trips the GFI breaker...every time. Tried some time ago to troubleshoot that with no success. It always works find at the pedistal at RV parks.
One other thing I "discovered" this morning, quite by accident: The fridge has not worked on propane for some time--no sparking or ignition. Today while I had the batteries disconnected from the trailer, I plugged in the shore power to 120V and heard the fridge sparking, and apparently firing up on propane. But later when I reconnected the batteries and plugged into shore power, no sparking. The fridge problem is not a big deal as it works on 120 and 12V, and we rarely boondock. But this behavior may be related to the electric issues...
Thanks again for the feedback.
