Having been in the same situation this is my advice:
First of all go through all your connections and make sure they are all clean and tight. Use battery anti corrosion
spray or whatever on all connections. That is, remove all wires and cables
and reinstall. Look for problem wiring
as you do this.
It is correct, your charger should be capable of carrying the load alone.
Those batteries are old and you don't know how they could have been abused, so you are going to be replacing them anyway...but the last thing you want is a bad charger to ruin the new batteries.
I would replace the charger. New one is not expensive. Look at defender.com. At worse case you have a backup charger.
I have the pro mariner pro tournament 30 amp. Yes its about 400 dollars but it has been very dependable.
If you have solar and just do overnights you might get away with a good quality automatic car charger or two (blasphemy, i know).
The solar can top up your batteries. Electrons is electrons.
I always carry an automotive charger with me. Yes I know they are not "smart" but its a cheap emergency
backup. Plus, if your batteries get severely discharged, some smart chargers won't start charging till you bring up the voltage.
Try equalizing your house bank. It might extend you.
Calder's 3rd edition on boat electrical
and mechanical systems makes a very good case for only having 1 house bank, not two. I would combine the banks when you install new house batteries.
I would also, when the time comes, replace the deep cell engine batteries with starting type batteries. You should be able to find out how many cold cranking amps are specified for your engine. As for me, I get the biggest that will fit, at Walmart or costco. At least 50% more than required in CCAs. The price
differential is nominal.
Why can't things remain where i carelessly left them?