Double check none of your DC jack lugs have continuity
The battery +9v and top +9v should only have continuity until you plug in your adaptor, check once you plug in your adaptor there’s no continuity between the battery power and main power terminals on the DC jack
There is no continuity between the battery +9v and DC/top +9v when the adapter is not plugged in, with battery used. But there is continuity between them as soon as I plug in the adapter, so they must be touching each other somehow.
My old eyes seem to see that the battery positive is connected to the +9 terminal on the power jack instead of the middle battery lug. Reverse the two red wires and it should work if I am correct.