I like both of these ideas.
This is a good conversation starter:
. . . you could also take your name tag out of the plastic thingy and, with your handy felt tip pen, add "soon to be at Department, College/University."
This will help people get in touch with you:
It would indeed be fine to say "Contact after July 1:" Big Box State U., BB City, Bigbox, 12345, USA"
....congratulations!
+ one more!
ps: It's not too early to start finding out from you current place if it is possible to have your email forwarded or even rerouted, and from your new place, how early you can set up a new email account. When you're in the midst of a move you may not want to think about such things.