Air Cargo is an option.
This is SouthWest Airlines Cargo page.
Southwest Air Cargo - Cargo Rates and Rate Calculator
Most airlines and some other carries fly out of Air Cargo as well, so check what flys from where you are to where it's going, and of course the flight options.
Be sure to read the airlines shipping guidelines too. SWA is 90 total inches (H+L+W).
I could ship my bike from here in Phoenix to New York right now and have it there tonight for $112.46
Or San Fransisco for $72!
For obvious reasons there can't be any fuel in it.
If it even hints of fuel they won't take it, so if you know you are going to being shipping it use something other than the bikes gas tank for test rides, thoroughly clean the carb, then put your ship ready fuel line on.
To pack it, your local bike stores and department stores should have bike shipping boxes if you don't have a couple yourself from ordered bikes, but you have to make one from two boxes unless you can find one big enough.
Basically you pull the handlebars off from the head with all the controls intact on it, and the front wheel off too, and maybe the seat and pedals if need be and zip-tie them to the frame to get overall size down.
Build a box around it and take it to the airports Air Cargo center.
If you can't get it to fit in one box you can always put the seat, gas tank and pedals in a second box.
The Air Cargo terminal is not the Passenger Terminal of the airport, it is a completely separate area.
You check it there and it goes in special Air Cargo bins and on the flight you choose.
When it arrives at the other end, the recipient goes to their airports air cargo center and picks it up.
I haven't shipped an assembled motorized bicycle yet, but I have used air cargo many a time, mostly back in my computer days when a part was needed ASAP.
The reason it's so cheap is it's easy for them to handle and it doesn't go on a delivery truck or through the passenger terminal as a separate item.
Hehehe... Can you imagine what it would cost you to take your motorized bicycle as checked luggage these day?
Jezzz, a bag of golf clubs is more.
Ohhh, one more note, the box will be opened and inspected so don't go crazy with the packaging before you go, and take your tape and banding tool with you.
Hope that helps ;-}