When I moved across the country, I used this forum to gather some info.
http://www.movingscam.com/forum/
There are a lot of scams out there. and the price variation between a good company vs a no name company is ridiculous. If you have enough stuff to fill a pod, that might be the way to go.
If I were the OP, I would consider renting a small van to haul your goods. Have your husband drive the rental, and you drive the Corolla. If you insist on riding with your husband, then you may have a friend or family member that would like to take a long drive. Than you could fly this friend back home and give them a few bucks for their trouble.
Yosemite
I agree with @Yosemite with renting a small truck and driving the Corolla. Simplest way to go and the added gas cost wouldn’t be that much.
“My husband and I are moving from Minnesota to Seattle in four months.”
“Corolla’s can’t haul much so adding a trailer hitch to my leased vehicle seemed pointless.”
“We don’t want to spend over a grand shipping anything.”
“I also read briefly that top racks can only withstand about 125 lbs of weight… that obviously won’t be worth it if that’s the case.”
“Basically, we aren’t really wanting to rent a uhal and drive separate, and we dont want to ship anything either.”
@CorollatravlerMNtoWA
Basically, you are moving and taking some stuff, comfortably, and practically without cost, but have eliminated almost the means to do it.
You Need to move, but have Wants that are causing problems.
You’ve painted yourself into a corner.
I think that either you’ll have to loosen your grasp on some more money, cancel the move, or make some other concessions. You’ve got lots of great suggestions here on how to make it happen.
CSA
@csa You’ve articulated the situation nicely. A $1000 limit on moving expenses is wholly artificial. My approach would be to sell everything not needed in Seattle, pack light small stuff in the car with a roof carrier (as we did), and pack the rest and have the box moved by a reputable mover.
Minnesota to Seattle is hardly a world trip and won’t break the bank. The cost of the movers can be spread over a period of time; I would borrow from relatives if I lacked credit and did not have more than $1000.
Keep in mind that used furniture is very cheap; there are garage sales in my area every weekend and I could furnish a 2 bedroom apartment for $1000 easily by shopping there.
Beer Budget, Champagne Taste. 
CSA
I have a champagne budget and beer taste. Lucky me!
"I have a champagne budget and beer taste. Lucky me! "
I have a beer budget and beer taste! Lucky me!
Did somebody say beer? OK, fine, I’ll have another! 
CSA
We are lucky guys, CSA, and fortunate enough to know it.
Ship your books to the post office in Wa. with the fourth class rate. Ship whatever you were putting on the roof rack and abandon the rest. Nothing is worth keeping if movers are involved.
Here is the cheapest solution. It won’t be fun, and some (or many) might consider wreckless, but it’s definitely the cheapest. Don’t blame me if you get stuck half way there. O.K., with the disclaimer out of the way…
I would put a trailer hitch on the Corolla and buy a 4x6 trailer. Sell the trailer when you get to Seattle. You may lose a couple of hundred dollars in the transaction. The trailer is small enough to see around, light enough (barely) for Corolla to tow. Just be careful not to work the transmission too hard. Make sure not to overload the trailer. Corolla is so light, you might get fish-tailing, if the trailer gets too heavy. Also, you will know when the transmission is working too hard when the engine rpm jumps up without down shifting. (That is when the torque converter clutch disengages, and transmission fluid temperature begins to rise significantly.)
I towed a 6x10 tralier behind my 2003 Crown Victoria from Oregon to Boston. It was manageable. I sold the trailer for $100 more than what I paid for in Oregon.
Well first problem is the Corolla is not going to be able to pull the trailer all the way without wrecking it. The second problem is that everyone in Seattle drives a Prius that won’t be able to use a trailer so no resale value. And if this is the cheese guy from Minnesota that I think it is, the car will already be fairly weighted down.
Third problem, I think adding a hitch would put a heavy penalty on the lease.