We are flying in to Raleigh in July and we would like to spend 4 nights west of Asheville near the Smoky Mountains, and then spend 3 or 4 nights in the Outer Banks. Any suggestions for lodgings that are scenic and comfortable, yet affordable?

The South Carolina beaches would be closer to Asheville than the Outer Banks. You could drive to Charleston in four hours but it would take you eight to get to the Outer Banks.

Lodging in Asheville would be The Grove Park Inn. I would pick Wild Dunes for great beaches.

MohleTraveler, would you be happy going to Wrightsville Beach/Wilmington (about 5 hours from the mountains) rather than Outer Banks? It's a beautiful beach, and Wilmington is a neat place to visit. Where you sleep depends on what you want to spend ... The Grove Park Inn is lovely but more than I usually want to pay (you mentioned affordable ... be more specific); I like the Residence Inn near Biltmore, has an indoor pool and is immaculate.

RDU - Smoky Mountains - Outerbanks - RDU is a HUGE amount of driving in one week. I agree with above - try to satisfy your beach/mountains combo with less driving.

If you want to stay west of Asheville, check out Maggie Valley, a somewhat small tourist spot near Cherokee, NC. There are several motels and cabins, many with wonderful views. There are still many independent motels in the area with a lot more personality than the chains.
http://www.maggievalley.org/
Since the casino came to Cherokee, the town has lost most of its charm, IMO, and I wouldn't stay there (unless you like casinos ...)

If you want to limit the driving, head to the NC High Country instead of to the Smoky Mountains (it's a couple hours closer to RDU). Blowing Rock, Banner Elk, & Beech Mtn. are towns to consider, and you're within an hour of Asheville if you need civilization.
http://www.highcountryhost.com/