I've now tried powering the multimeter directly from a lab power supply, just touching probes to the PCB pads that normally contact the battery terminals in the rear part of the case. I still got the same result, beeping, so that should rule out the batteries/contacts causing the issue.
I don't think I can remove the test lead terminals without desoldering but I've looked inside with a magnifying glass and light and can't see anything more than some dust that could short out the terminals. I've also tested continuity between the 2 PCB pads of each of the terminals. The V and COM both have continuity with and without leads plugged in. The A and mA both have continuity only when leads ore plugged in, otherwise not.
I've also inspected the PCB more closely with a magnifying glass. There was muck on the rear side of the PCB (nearest the batteries) that looked like some residue from something, the front side was all clean.
There is corrosion on the screw and PCB ground via that holds on a shield over the main processor and holds the PCB to the front part of the case (there's one other screw lower down that was fine).
There is corrosion on the left-most pad of a long, thin, vertical white component.
And there is possibly some corrosion on one of the traces leading to the main processor; or it could be something else.
These are circled in the attached image.
What is the white component? (I think it's the thick film resistor network?) Should I clean and re-solder the corroded joint?
Any other ideas?