If you really want to get to the root of the issue, you are going to have to do some work !
First, get a 6V charger and charge each battery individually. This does not have to be a fancy multi-stage charger. Just leave each battery on the charge over night.
Second, label the batteries. Immediately after charging, check the voltage. WRITE IT DOWN. After that let each battery "rest", unattached to anything, for 24 hours. The voltages after charging and after rest needs to be about +/- 0.1V. If any are lower, they are on their way out.
Third, carefully CLEAN and inspect all cables and connections. Replace any that are suspect.
Fourth, get an amp-clamp meter. Many to chose from on Amazon for < $50. (I have a Uni-T UT210E.) You may want to consider installing a voltmeter and ampmeter (requires installing a shunt) for more accurate readings. Under $20 from Amazon.
If you are consuming more than about 0.100A when everything is off, you have a problem somewhere else.
Last, run some heavy load with no shore power so that you purposely discharge the batteries at least 10%. Check the voltage AT THE CONVERTER/CHARGE AND AT THE BATTERIES. Plug in to shore power and check those two voltages again. Anything more than about 0.100V and you have a cableing problem.