Car lights work on reverse polarity so a ground will energize them. From the rain looks like something got wet and is grounding the circuit energizing the light. Other than suspect water intrustion at the SJB.
I had the same issue after some rainy days and it ended up being the fuse box behind the passenger kick panel getting soaked, I disconnected the battery and cleaned out all the contacts with electric clean and then let dry overnight and I got lucky everything went back to normal the following morning good luck.
Mine just started staying on after I turn. I can turn it off manually it just doesn't automatically turn off when the wheel is returned to center. Oh and only when signaling right. I suppose I need a new switch/lever thingy.