Interesting that it slowly counts up on Voltage and down on Ohms. I don't really know what's wrong with your meter but I'll throw out some suggestions for your consideration.
Start with a good cleaning. Take off the LCD and dunk the board in IPA. Use a brush to scrub the board. Pay particular attention to the areas in-between the leads of the analog processor chip U1. Allow to dry thoroughly, a fan and/or gentle heat is good. This procedure very often fixes or at least improves weirdness in the front end. It only takes miniscule leakage currents to cause symptoms.
If you haven't already, check the PTC thermistor RT1 and the big fail-safe resistor R1 for correct values. The thermistor should be around 1.1K. In voltage mode the input goes IN through these series components. In Ohms/Diode/Cont modes, the test current comes OUT through these components. I don't have a schematic for this particular model so I'm not sure what R1 is on yours, but it is probably either a 3.5K or 2.0K wirewound, or maybe a 1K film.
Check to see if the ADC input can actually respond at normal speed if driven directly. Find pin 2 of the voltage divider resistor network Z1. (Pin 1 is on the lower end and has a square pad.) Set the unit for DCV. Apply 200mV between the COM jack and Z1 pin 2. (Careful, do not overload this input.) The meter should respond by quickly displaying "2.000".