Using shore power with bad batteries

As said above, maybe only one battery is bad. Just remove the bad one.

Yes, just disconnect the batts (wires off neg post, not by using disconnect switch)and the converter will do all the "12v" work.

By leaving the positive post wires on, the slides , jacks etc that are battery direct will still have a path for the converter to run them. If you remove the batts entirely, you can clamp those pos wire terminals together (eg vice grip pliers) and get the path, but be careful not to let the clamp touch the frame (big sparks!)

If you leave soon to be dead batts on the converter, the converter might get overworked trying to charge them (not too sure about that, but why take a chance?)

Are they both 12v batteries? You can always charge them separately and then only use the battery that reaches full voltage until you can replace them. :C

Depends why they are low. If a cell is bad or shorted, it is possible to catch fire.

But even if it is safe, it could be a strain on your converter/charger.

I would rather have a battery that charges up to full voltage and dies quickly than have one that won't reach full charge.

That's my opinion ;)