My Sony A7R5, 24-70 and I got thoroughly drenched about a year ago when I was just coming to the end of shooting a day to night time lapse, so it was dark and I didn’t see it coming. It was about an hour before I could dry it off properly but it was all fine. Also, I shot all day in the rain in Rotorua with my RX100III, which I don’t believe is weather resistant. It was quite warm and steamy and I protected the camera as best I could but it got quite wet. It was also some time before I could dry it properly but it was fine also.

Are you able to tell me how you setup a button to turn off the screen? 

I have an A7RV and I have done night time lapses. I just have a button set so I can turn the screen on and off. I don't have the problems you are describing. Maybe a full camera reset may help. I don't know.