Contact Dave Fountain at Executive Motor Coach. I have used him twice to ship my car from CA to the East Coast and back. He is well known in the Mopar community. He can ship enclosed trailer. I would use no one else.
Enclosed trailer might be the way to go if not too expensive. My 124 was shipped 2,000 miles on an open trailer. It had difficult-to-remove grease streaks all over it and the engine bay was covered in dust.
I've shipped numerous cars & bikes all around the county and there are many reputable choices, but I have always used the enclosed method since there are too many things that can happen when using open transport unless you are going short distances. If you use "space available" or "stand-by" it will save you money over selecting a specific pickup and drop off time. Also, if there are major auctions or car shows the good carriers will be booked so that is one way to tell if you are getting a good carrier. But based on personal experience I would recommend Exotic, Reliable, Passport or Horseless Carriage. I would advise getting quotes and time frames from those 4 and pick the one that fits your budget and schedule.