Please keep your itinerary posted. We are not far from the southern end of Shenandoah National Park where the Blue Ridge Parkway begins. Waynesboro, VA, is just a short drive off the parkway there and has many dining and lodging options. There are a couple of cheap motels near the BRP at Fancy Gap near the southern end of VA as well. Don't expect anything fancy at Fancy Gap! Ask me how I know. 😜
Once you get into North Carolina there are great options at Boone, Little Switzerland, Asheville, Maggie Valley and Cherokee but inexpensive will be hard to find in those areas. I used to travel the BRP a lot on motorcycle. I'd think in a Model A, three days to travel the 469 mile length would be comfortable. With stops you'll only average about 25 mph.