I know regular rv refrigerators have an internal switch that will go from 110v to 12v automatically if the 110 drops so it keeps on running. I have a 12v cooler/refrigerator and looking for an item that will mimic the dual power that so I can just plug the cooler into a 12v slot of a "box" that will switch it so when I am on shore power, it runs off the 110v instead of the 12v without me having change plugs. If you know what this "box" would be to buy, let me know.
If your rig has a 120VAC-12VDC converter (i.e. it can charge the batteries from 120VAC) then you don't need to do anything: plug into 12V and when you are on shore power the converter will provide the 12VDC.
Yep, A standard RV power converter will do it. There are lots of newer options though and some go by different names. Many inverters nowadays have this capability (3 stage or multi-stage DC charger when on shore power), some are called powerverters.
I nominate Scalf77 for the best explanation and advice on this. Hopefully he will chime in.