This is a bit of a headache, but you'll get a listing of national and regional movers. I used Movers.com to get the list of movers (and most of them contact you). You end up with quotes and then look them up at the Better Business Bureau and make sure that they are accredited.
I'm moving SF Bay Area to Chicago, and got quoted $1400/3000lbs, with free elevators, months storage, etc.
Things to keep in mind:
storage fee
storage handling fee (if they have to put your items into storage)
Shuttle fee - if they have to rent a smaller vehicle to transport your items (this is likely in downtown Chicago, but probably not in Evanston)
Elevator fee
Stairs fee
Long distance haul fee
Packing fee
It's annoying, but make sure you get all of the fees penned into the contract (in particular if the fees are zero dollars) so you don't get taken for a ride.
Good luck!