Hi Christy,
Sorry for the delayed follow-up.
I did play with it a bit and I do agree (and appreciate) with all responses but I think I can add something...
It seems that the basic fact of having a "G Follow" on every page already covers the rel=publisher part and the Google Structured Data Tool @Adam here recommended does show my company as the publisher even though I did not specifically add rel="publisher" (simply because of the Follow button).
This is also why I don't know if I need it along rel="author" since I have the "Follow" button on all pages but I assume that you are correct and the two are not needed together.
Thanks