While performing training exercise in my backyard I can feel my shotgun noticeably lean to the left. But when I dry fire and quickly shoulder it, the gun is pointed to the right of where I'm looking. It naturally wants to aim right so if I pattern it I reckon it would print right. Is my barrel off center? How can I check this?

I was thinking of moving the light to the right side to help balance the weight.

Your mount is bad, possibly due to a combination of improper mechanics and a poor gun fit.

Get your shotgun fitted by someone who knows what they are doing.  It will at least address the mechanics of your mount, and will also help in deciding what stock adjustments are needed.