I 90 Mountain Passes

The mountanous part of this trip is all at the beginning. - west of Spokane in the mountains of Montana to Bozeman.   The rest of the way across South Dakota and to Rochester (Minnesota) is pretty flat.
The alternate trip would be using I-80 from Walla Walla down to Ogden (which still has some mountains) and then across the plains to Chicago
I would make my decision depending on the weather at the time, and the advice of people there in Walla Walla as to which route to take, and try not to be tied to a schedule which keeps you from waiting for better weather.

Always people die on icy roads in March and April. It is when you brake on ice that it becomes a problem. So do not slam on brake. It is what you hit, or get hit by, that becomes the issues. You will need blankets, candles, flashlights, reflective vest, water, food, gloves, hats, coat in the car. As soon as you are near ice, find a big parking lot with no poles and see how your car spins out on ice. Stop driving at 4.