You may have done this, but since you didn't mention it specifically, have you checked the Snapping toolbar to see what snapping options are checked? In particular, in the dropdown arrow, make sure Use Snapping is checked. You can also go to Options on the same dropdown and choose to snap to basemap or feature service layers as well.
You also mention you're georeferencing. Snapping doesn't work on the image file itself, only Arc features (lines, polygons, points, etc). Again, you probably know this, just thought I'd mention it.
