If this is about the sun moving because of not being centered, that doesn't make any sense at all to me.

Most of the wobble would be Jupiter, so shouldn't Jupiter's orbit be the approximate period of the wobble? Which is 11 years, which is obviously way too short to be in a discussion about global warming.

I found a picture on Wikipedia that seems to confirm that the barycenter takes about a decade to go from near the center of the sun to roughly the furthest away that it gets.

https://upload.wikimedia.org/wikipedia/commons/0/0c/Solar_sy...