I'm cross eye dominant also and have no issues with the rifle in my right hand and sighting with a telescopic sight or irons with the dominant eye closed. With a red dots & Holo sights I keep both eyes open and have no problems. With pistol I have it my right hand and sight with both eyes open but with my left down the sights.
All you have to do is shoot a lot.
Airsoft, where nothing is hurt but feelings.