Hi there,
I've come across this issue in the last couple of months. It has come and gone once or two times. I don't know what I've done differently to make it work/not work other than come back a week later to try again.
I went into an existing Field Maps (that already has offline areas created using the polygon tool), but when I went to click on it to create a new offline area (as a polygon, not rectangle), I could see and click on the option to create the areas as a polygon, but when I click on it nothing happens. I'm not able to draw in the polygon. When I click on the rectangle option I am able to create an offline area.

I have asked our local ESRI support office. They were able to replicate it in my map, but not in any new ones and haven't heard of this issue elsewhere.
I can't think of anything I'm doing different to anyone else except for I'm working in a different time zone than the client at the moment.