Multimeter started giving gibberish readings

Usually, the rotary switch has started to develop contact problems. Sometimes "working it" does make things better, for while. Second typical cause is broken input jacks and/or broken test leads.
But of course it could anything else. I assume you replaced the battery with a new one and used a 4mm banana jumper lead for a 0Ohms test, if only for double checking?
 
Test leads themselves can break internally, as they get a lot of flexing in use, but yes connectors can go slack and no longer make reliable contact, and the multi-way rotary switches in multimeters eventually wear through their gold-plating and become less reliable.
If taking a resistance reading, then its worth checking with the probes-shorted reading to check if everything's OK - typically you'll see a value between 0.2 to 1.0 ohms. The only way to get accurate low-resistance measurements is with a 4-terminal system (some bench multimeters do this, and its a lot more tolerant to contact/lead issues too.
 
A blown fuse would result in a permanent OL display. If the leads were defect, it would also happen in other modes too. So both are unlikely.

The rotary switch sounds reasonable, maybe some dirt or fluid found its way in it. It could also be defect of a AD circuit if the leads got exposed to high voltage, especially if it was HF, it can bypass protection measurements or can direct induce into circuit traces. Most MM aren't hf shielded.
 
The MM should show low battery then. But yes, that's also possible and a good idea to do that. The resistance measures draw the highest current, after all.

E: I feel really stupid I had not thought about that myself first.