The following should do ...
Well , I also haven't changed info. on the SSN yet (PP stamped on Aug 16) but would prob. go next week.
I don't think it should be an issue about when one does it - just that we should remove the restriction on the SS card, the earlier the better. And yes, you can go to a local SS office.
I'm assuming that the following docs should be fine - SS card, Passport (showing I-551 stamp), a photo-id.
Hope this helps.