Not at all cross-dominant, and unfortunately I was an avid golfer [100+ rounds a year] for 30-some years before I made the connection. When I putt with both eyes open, sometimes I am not lined up properly with the hole. Being right-handed, if I close my left eye when lining up to putt, I am always looking down the correct line.
That said, I also was an avid trap and skeet shooter for many years, and extremely nearsighted in the right eye [since corrected by lasik]. I broke many 100-straights in both trap and skeet shooting right-handed but looking out of my left eye, with my right eye closed. Somehow I must have made the appropriate adjustments, because with my left eye only, I’m always looking well to the left of where my dominant eye points. However, I was money on hard lefts out of the house, ‘cause I always had an extra built-in lead to the left. Hard rights were my downfall.