The old battery powered some lights and a pump and was charged by the converter. If you connect the new 12 volt (6 volt pair) to the same spot then the old stuff should still work. If you get a shaded camp spot that has power you can plug in and set the switch to the convert position. Otherwise leave the switch in the battery position.
With emphasis, connect panels to controller last. Connect controller directly to battery not directly to converter. The two charging sources can coexist just fine.
There is the possibility that the converter is broken and will put a load on the battery. Use a meter and measure the not charging in the dark battery current. One light on, other light on, both lights off. The lights should be an amp or two, both off should be zero amps.