Your 110 AC powered lights are not compatible with 12 volt appliances. Unless your boat has an inverter, or "smart" charger, that converts AC to DC then my guess is that your 12 volt instruments are simply running off the battery.
So from that point it is just the simple task of educating yourself on how 12 volt systems work. Pretty simple
if you acquire a copy of
"the 12 volt Bible" by Miner Brotherton.
Every boat owner should have this basic, easy to understand book in their boat library. All answers will be there.
(just a note, in the early days of Catalina yachts, ordersheets were the norm, which means almost all functional equipment. From anchoring ground tackle, bilge pumps, stoves, Shore power systems, which are the 110 v AC circuits, etc. In most cases there was only one battery, which was usually charged by the motor's alternator, and provided a source for running lights and a few instruments, maybe a couple of interior lights..but litte else. These systems could all be upgraded on the order sheet so they were installed an operable when the boat was delivered. People were pretty thrifty back in those days. But eventually Catalina and the other production builders included these systems as standard. It's importatant for you to research your boat's systems so you can modernize it got comfort and performance)