I have never seen a DC jack that disconnects something and replaces it with something else. They usually just disconnect a non-rechargeable battery when the adapter is plugged in.
I'm not entirely sure about this as I only learnt about it yesterday here: https://www.electro-tech-online.com/threads/automatic-switch-from-battery-to-ac-dc.85737/ but as I understand you get "switching" dc sockets like this one here: **broken link removed** which have three terminals you can connect to a 12v component and a battery. The positive terminal for the battery "breaks" when an adapter is plugged in, switching the 12v component over to ac-dc power. This is the sort of socket I have put the diagram. I went over the idea in the thread linked above and was told it was alright.
I'm pretty much satisfied now ... but just in case anyone would like to continue the discussion - it would be even better if I could wire everything up without a switch involved. (I won't hold it against anybody if no one can be othered )