Actually, I was pushing both options.
If Amtrak had a workable train with roll on/roll off service, you could get there without having to disassemble your bike. Sure it might be as slow as driving, and not a fiscal bargain, but it is an option I keep trying to explore on my trips. Closest I ever came to shipping by Amtrak was an Amtrak bus ($10 for unlimited use of bay #2).
Both Amtrak and Greyhound shipping are worth looking into and you would not be bound by the usual box limits that go with shipping via DHL, FedEx, or UPS. Greyhound can probably get from anywhere to anywhere else, whereas Amtrak might require you to drive to Chicago in order to ship to Tuscon.
Just remember that the bike store box will probably be considered oversize by DHL, FedEx, or UPS, and still be expensive to ship. USPS might be more forgiving. If you think you are going to do a lot of traveling, then one of the commercial cases might pay for itself. In shipping a bicycle, it really is all about boxes' length + width + height, and then its weight.