Let's see...
I'm LH/LE (left-handed/left-eyed,) although I head most lefties tend to be LH/RE, or cross-dominant. I predominantly shoot left-handed with rifle (but can switch with shotgun) and can shoot ambi with a pistol.
I will usually aim with the eye I'm "shooting" with (LH/LE and RH/RE,) but it's taken over 20 years to get there. I tend to shoot with both eyes open, or with the "off" eye squinted (but not shut.) It is possible to shoot a scope with both eyes open, and it WILL take a good deal of practise.
Knowing your dominant eye is helpful, but it causes no trouble. Also, wearing contacts should cause no trouble - I go back and forth between contacts and glasses often, with no real changes in my shooting.
It is going to be an issue in practise - practise often. Once you get it together, you just need to maintain it. Have patience.
FFZ