Unused batteries discharging; is solar trickle the answer?

All 3 of the previous answers are correct in their own right.
You just need to pick the one that best suits you.

And it looks like your present batteries are toast.

At a minimum you need new batteries and a multi-meter to test them and your converter.

P.S. If the battery disconnect switch is left in the OFF position, your 12 V stuff will work on shore power or gen but the batteries will never charge.