Strange coincidence, I was just talking to someone at the FAA about
this today. Basically as Martin said, you have to go to a FSDO in
person to do this.

The paper licenses are no longer usable after 31 March, but they don't
"expire". By this, I mean that you can still update the paper license
to the new plastic license after that date, you just can't fly with
it..

Well before your next trip to the US, go to the FAA website above and
start the paperwork so that you have it all in place before you get to
the US. Make an appointment at the closest FSDO and they can issue a
temporary that you may fly with and the plastic license will come in
the mail within 90 (?) days.