I'm trying to get the iron sights on my Ruger 10/22 zeroed at 50 yards. Elevation wasn't a problem, but it was shooting to the right so I took a punch and tapped the rear sight to the right and It's still hitting in almost in the same place.

What would y'all recommend? The only thing I can think of is tapping the front sight to the left because I don't think it would be a good idea to move the rear sight anymore.

what's your finger placement on the trigger? finger placement and shooting

If you're hitting right of your target the rear sight needs to go to the left. Or you can drift the front post to the right.

It helps me to think in exaggerated terms when deciding to move my sights. If I move the sight a lot how will that change the way the barrel moves when I align the sights? In which direction?

Once you establish a good 50 yard zero you can apply wicking grade Loctite to the sight dovetails. This will insure that they stay put.