You really shouldn't focus on shifting your weight back at all. A centered pivot doesn't have much weight shifting back. The pressure shifts back, but the weight won't (your arms will transfer weight slightly).
I read that bit about the "weight shift" above and wanted to comment on it, so I'm glad you said it again.
Don't worry about it for now. The hips are fine (they might be too far forward, but not getting them forward with the head movement would be way worse right now).
Practice in a doorway.
Top: Yeah, I noticed the sway. It's almost certainly from trying think about shifting my weight. I'm guessing that it's not something I should even think about at this point unless I can avoid drifting back. It just really feels like I keep all my weight forward right now when I focus on keeping on the wall.
Bottom: I honestly didn't even notice/think about how far forward my head was getting. I won't worry about it now, but it's interesting to see how much it moved. Are my hips getting too far forward also then? If so, would this be partially due to how I drifted back?
Thanks for the feedback, I'll try to do some more work tomorrow. When I do I'll really focus on not swaying back on the turn. I really need to get the mat so I can practice inside on days when it's so stupid hot or when it rains.