It doesn't matter what IP you log in on. The only requirement is that you register as multis with other players who share your IP. If you're just moving by yourself and aren't moving in with another player, it does not apply to you.
If you currently have a multi and won't once you've moved e-mail the multi director to unregister.
If you will have a multi once you've moved you need to e-mail the multi director to register.
If you don't have anything to do with multi-accounts the only part of SWC that cares what your IP address is presumably the exempt list a few people are added to if they have certain problems with the site. I believe that would need to be updated manually, if you aren't on that list then you don't need to do anything.