If you have the time, I'd try and find the sections on the DMM, the parts and pins associated with the diode/cap mode. I would do it in something like LTSpice, and then it's easy to see to sometimes if removing 1 resistor or other part, or shorting some cap or diode, etc, could double or halve some current or voltage.
If you're lucky, the problem is outside the main chip. Here's the datasheet for the IC, there's probably schematics out there for full DMM's with the IC or something similar. Or might be bigger datasheets with more design guides.
I think a lot of diode modes work like resistance modes, by sending out (or sinking) some reference current, like 2mA, and then measuring some voltage drop. And they have some parallel resistance with the current source, to clamp it at 4V or whatever, when the probes are open circuit.
If you have another DMM, measure the output I, V, or you could make some circuit (if you have the parts and some know how), to get a rough idea of the current and voltage from the DMM.