I am new at this and have the same problem with 3 ac/dc transistor radios. I am not following exactly what contacts are being shorted out. Is it the + and - contacts on the AC? I marked up photo with arrows would take the guesswork out for me.
Yes. Try shorting the switch pins on the socket.I would say its the dc jack socket thats faulty, most mains/ battery radios switch off the battery positive lead via the mains lead jack socket.
Post pictures please

Some AC plug in's have an auxiliary contact that disconnects the battery. Have you tried unplugging the AC connector?Repairing portable radio, AC works, batt doesn't. I have 2 radios from the 1960s-70s. Both are VISCOUNT 1860.
https://www.radiomuseum.org/r/viscount_18_transistor_acdc_afc_6_band_1860_japan_304.html
It has a direct AC plug-in with a two-prong connection. The AC works fine but the batteries do not work at all. It has an AC/DC switch, but switching it does not help. Both radios have the exact same issue.View attachment 190113