Recently my 3-speed ceiling fan, which does NOT have a remote, won't turn off and only runs on high and low.

What make is it? How is it controlled?

It is a Hunter controlled by chains.

Did it just suddenly stop working correctly? Or has it been like this since installation?

No off position implies a broken pullchain or someone messing with the internal wiring.

It just started doing this yesterday. We've had a lot of rain... could the humidity affect it?

Did it start with the problem as you pulled the chain? No off position sounds like a bad pullchain

Ceiling fan pull chain replacement and repair - Ceiling Fans N More