I was faced with a similar issue on my HP3466A but spraying contact cleaner in through the front of the switches while holding the board vertical was enough to solve it. Maybe someday I will need to take them apart, but for now it's OK. It is absolutely true that the bad contacts will cause the unit to switch to resistance mode.
Fun story on mine, I ordered a vintage & broken Keithley DMM and the company sent me the HP3466A first by mistake. Then they sent out the correct item and told me to just keep the HP because it was faulty. But the only major issues I found were the intermittent buttons and of course bad batteries. It wasn't a vintage DMM on my radar at the time, but I enjoy having it now.
The LED displays are always cool...