Welcome to the Forums. All of the problems that you describe are based on a lack of 12VDC contrl voltage. That comes from 2 sources: your bateries and the Univolt that should keep them charged while connected to shore power. It seems that the Uni is not doing it's job. You can test the output with a multi meter to see if it is putting out 12VDC+ at it's output terminals.
That said and realizing that they don't do a proper job of charging batteries in the first place (IMHO), I would replace it ASAP. For a replacement, I would be sure to get a converter that has 3-stage battery charging like the WFCO or Iota brands. They can be purchased complete with a new fuse panel for all of the wires that are associated with your Univolt.
You should be able to do the replacement yourself, and you will find several threads dealing with the subject in the 'search' section of the Forum. Have fun!
Lew Farber...RVIA Certified Master Tech...ABYC Certified Master Marine Electrician
SOLAR TECH ENERGY SYSTEMS, INC.
AM SOLAR ...MAGNUM…VICTRON ENERGY...LITHIUM & LIFELINE BATTERIES…MARINE REFRIGERATION