How exciting! I envy you.

I haven't used Bike Flights myself, but I gather it is the answer to your question. Amtrak will ship a bike too, almost intact (pedals off, handlebar turned sideways). I've done this and it was easy; I rode to the station, walked away without the bike. If you don't have a convenient Amtrak station with baggage handling facility, never mind. Bus systems like Greyhound offer a similar service, I used it many years ago, but again it may or not be convenient to where you are.

Either way, all best wishes on your tour! i hope you'll update this thread with a "wish you were here" post now and then.