L.-
My son also started having trouble with breast feeding around 3 weeks old. He nursed great until then, but at 3 weeks he started gagging and crying and basically latching on and then coming off. At one point, I also thought it happened more often on one side than the other, but after time went by, I realized it was just a coincidence. My lactation consultant was great. We figured out that he was getting too much milk at once and couldn't handle it. To slow down your milk, you can gently press down on one area of the breast (always switch the area so you don't get a clogged duct), or you can lean back if sitting up or lie down while feeding him. My son would still gag, but it happened less often. The gagging pretty much disappeared after he hit two months. As for the latching on and off with crying in between, that seemed to be his tummy bothering him. Even now, about to be 4 months old, he still does this sometimes. Now, I know to expect a poop or some gas to come out. After I change him, he'll usually eat some more. If you have access to a lactation consultant, they can help figure out what is going on in your situation.
Also, if this definitely only happens on your right side, you may want to have his ear checked just to make sure he's not sick.
Good luck!
-T.