An update on how I am accomplishing the move.
This morning I drove to SEA with the dog (37 pound mini Australian Shepherd) and the large cat (21 pounds) and shipped them as air cargo on AK's non-stop to DCA, where my daughter will pick them up and keep them at her apartment until I get there. The quoted fee was around $800, but when I arrived at the cargo warehouse, the computer made a mistake and spit out a fee of only $150! I pointed that out, and the staff made several phone calls, but ultimately decided it was much easier to leave the fee at $150 than to make the computer change.
Tomorrow night I'll take the redeye to IAD with 1 cat in the cabin with me, and 1 as checked baggage. I'll rent a car at IAD, my daughter will meet me there with the other two pets, and I will drive them all to North Carolina, our ultimate destination, but not do-able with non-stop flights, easy connections, or acceptable cargo shipping arrangements.
I have a temporary summer job, and the agency making my job placement arrangements is providing the car rental for the summer, which I will return to IAD at the end of the summer, then figure out how to get myself back to NC.
The whole thing felt like planning the Normandy invasion!