If you can't find radio broadcast archives you might at least look for newspaper archives. The newspapers that operated in the region might have published schedules for radio shows, giving you a more solid starting point as far as the duration of the broadcast years for the Carolina Barn Dance. Back in the day you would have had to sit down with microfiche and look at individual blocks of pages of specific newspapers published during weeks or a month, page by page. Now most of those are digitized and usually searchable (you will have to pay to use a database). Would the radio show have played only part of the year, or was it on every week all year long? Sample several months of the local and regional newspapers and you'll likely find a paper that carried radio broadcast schedules. You'll probably find them next to the TV schedules.

Another option is to look for universities in the area. Any of the larger state university libraries will have Special Collections (archives) of some sort and if you get a result in your search then look for Finding Aids to narrow down what is in a particular collection. To save you time I ran an advanced search in Google:

These results came from searching the exact phrase "Carolina Barn Dance" and all of the words "Spruce Pine, North Carolina". I see three solid results including a finding aid.

SRS (Retired university library professional)