Rent a motor home, load it up, get some help with driving if needed and enjoy the trip.
Six years ago we didn't move nearly that far, the move being only 3 hours distant, so I moved roughly 60 long arms and roughly the same about of handguns by just layering and padding the longarms in the back of my pickup with blankets and furniture bads. No way I had enough gun cases, socks, and such to cover the longarms. A large assortment of zippered bank deposit bags and a few zippered pistol cases protected most the individual handguns which were then packed in surplus 60mm ammo cans for the move.
Colts, Smith & Wessons, 19th/early 20th century Winchesters, U.S military stuff in "trash 'n treasure" condition all made the journey in fine style with no ill effects.
We had "help" in the form of well-meaning friends 30 years ago when we moved about 6 years after we married. Despite my cautions and best efforts, there are some guns around here that still bear the scars of that move and a P 38 turned up missing, never to be seen again.
If you are driving through then perahps a small covered trailer provided with ample padding would work for you. Be aware of any wetness that might intrude in the trailer if you drive though rain during the journey.
Welcome to Texas! We like folks to bring all their guns and come here.
When I used to take dozens of handguns to guns shows, I had zippered gun rugs for each handgun. I just threw the cased handguns in a duffle bag, which made for easy carry. (Real GI duffle bags are very sturdy, and have wide, strong shoulder straps, thus allowing one to throw a couple of bags at a time over each shoulder.) I would take the duffle bags into a motel room with me each night for security.
If you do not have enough gun rugs for all of your handguns, wrap them in bubble wrap. Put the cased and wrapped handguns in the duffle bag or bags. Take the bag or bags into the motel room when you stop at night. GI duffle bags will hold long guns as well, even if they stick out of the open end of the bag. (It might be advisable to bind several long guns into a bundle so that one does not shift and fall out of the bag.) If you do not have enough long gun cases, wrap them in bubble wrap as well. The flexible nature of the duffle bags and their contents will allow them to be "molded" into the space available in your vehicle. If you have some help each night, you can unload in one trip and take them into your motel room.
Put the ammunition boxes here and there into available space in your vehicle, making sure it can be hidden from view some way when you leave the vehicle.
First, welcome to Texas! Williamson county, as you probably know is just north of Travis County (Austin area). Williamson county is pretty conservative while Travis county is the liberal hot spot of Texas. Even after saying that, Travis county has most of the rest of the country beat on firearms rights.
I have some relatives in the Bellingham and Lynden areas of Washington.
I really don't have any new ideas about transporting your firearms, only to say I know you will be on pins and needles until they arrive safely in Texas. One poster mentioned selling your ammo, I would not do that as you won't get retail prices for your ammo and the prices just keep going up. For instance, I was buying Blazer aluminum cased .45 long colt a year ago or so for $39.00 per box, just checked the two days ago, the same ammo is $53.00 per box. And forget about finding .22 in large amounts.
I know it rains a lot in Seattle and generally stays cool all year. Just get ready for some 100 degree weather from around the end of June until the first of September. I think you will like living in Round Rock/Georgetown as you are just a stone toss to Austin and maybe 100 miles to San Antonio and just to the north about 150 miles is the Dallas/Ft Worth area. If you like hunting and fishing you will be in hog heaven.
Good luck on the move and I hope everything arrives safely and in the same condition as when they left Seattle.
Go with the motor home option. You can take as much time as you need. I moved from NY to Yuma a few years ago. The drive took me 52 hours with a 3 hr nap in OK. Everything arrived safely.
Welcome to Texas. The motorhome idea sounds good and if can enlist help driving that would be the way to go. If not, safely pack and cushion your firearms in duffel bags as suggested by Judge and stay at motels where you can pull right up to your room. Bring contents in, sleep, put contents back in car, repeat til you get here. My brother in law is a Constable down where you're moving though they live in Hutto. Beware of Austin traffic and welcome again to the Free Republic of Texas.
Welcome to Texas. Motorhome idea sounds like the best way. I live in north Austin area and I really enjoy it here. I can take you on a tour of local gun ranges when you get settled down.
It looks like you will be very welcome in Texas! Motorhome or large SUV sounds like the best option. I would love to move to Texas. But grandchildren are in NJ. Last time I visited was in 72.