I believe its actually a bit shorter to go to LA than to Seattle, although its close enough that if the specific starting and ending locations could switch it slightly.

Since it is not a significant difference, I'd probably choose LA as your shipping point. There is no way of knowing what weather you'll see at this point and the odds of seeing bad weather going to Seattle aren't really that much different than going to LA. However, by going to LA you'd have a few more options to try to avoid bad weather (I-80, I-70, I-40, and possibly even I-10). You also would be able to go south first, if they are seeing bad weather in the great lakes states. Just don't make the assumption that by going through "warmer" states, you'll avoid bad weather. They still see snow in all of those states, and often that warmer weather you speak of is just enough to turn a moderate snowfall into a severe ice storm.

With 8 days you have some flexability with your route options, and as Dave mentioned, you'd have the luxury of being able to sit and wait out a storm if you had too.