If flexing the jack a certain way solves the issue, then just keep doing that while you hunt?
While hunting, the DMM is off, so I can't "hunt" for real, just try to find non-connections where connections should be. I couldn't find any so far...? So flexing couldn't make it better... I am a bit confused about this.
Also, it looks like the sense connection for the mA/uA jack goes into an inner layer of the PCB then goes under the selector? Does pressing on the selector make any difference?
The rotating selector? No, nothing.