As above, 1000 miles of highway driving will have little effect on your car, as long as it has been serviced and is road worthy. If it were not in a condition to be able to complete a 1000 journey, then it's not really worth the expense of towing it across the country. For peace of mind it would be worth considering joining a motoring organisation, AAA for example. [Thats if you are not already a member]

Have a safe trip.