I've always been a sucker for anything British. I recall in my school days a really cool Morgan, right-hand drive and everything, in my US hometown. I loved that car and wanted just to sit in it. I think your car would be a hit...but I also think you should consider some of the ramfications of a right-hand-drive car in a left-hand-drive world. We face this problem (the other way) with our US car here in the UK.
- Drive through windows (just about everything in the US is drive-through nowadays)
- Machines that you have to talk through, or take the ticket out of in order to get a gate to open
- Passing cars on the road (your vision around the car in front is more limited)
- Windshield wipers clean the window area more for the driver and what the driver's view is needing to see based on what side of the road is being driven.
- People in other cars sitting at intersections (a four-way stop in the US, for instance) will be looking at the person they assume is sitting at the wheel for an indication they can or cannot proceed. If your passenger is looking down, the other driver will go ahead thinking the driver isn't paying attention...all the while you could be staring right at them.
But to finally getting around to answer your real question
...
Yes, you can ship your car. If you're also shipping household furnishings, the car will just go in another container with the rest of your stuff. There is some processing, but aside from the usual red-tape stuff, it's fairly painless.
We brought my car because it was already paid for, was my little baby, but...truth be known...the sale of it fell through at the very last minute. And we've been using it for the past six (more) years. (It's now 11.)