I would start a job search in both communities in which you're interested. Test out the market. Try looking on the Internet and perhaps subscribe to a local paper. Perhaps your family can help you with this since they live in the area. It's been much too many years since I looked for a job and so I don't have suggestions on how to do that.
I doubt that anyone would hold a job for you until summer but you can get a feel for the market. Find phone numbers for places you'd consider working and call their human resources people and ask questions.
Do the legwork as much as you can from where you live now.
You can rent, sell, or leave empty the house in IL. If you may come back to it, then rent it. Or if you don't have a job when you leave, leave it empty for a short period of time. I guess I wouldn't do anything with the house and not even move much until I had a job in the new place unless you know the move will be permanent.
When I was a child, my father went to the new state and found a job and a house while my mother and kids stayed in old house and state. Once he had a job, they put their old house on the mkt. and once it sold we all moved to the new state.
I suggest you could do something similar if you wait until schools out. Take your daughter to the new state, stay with friends/relatives and look for work. Once you find work, find a place to live and sell your house in IL. Before you leave IL engage a realtor who will then sell the house when you tell him to do so. I think you could have the realtor or a friend keep an eye on your house.