My first reaction (and I mean this sincerely, not trying to be offensive), is "Great!" You're learning first hand that your presumably comfortable lifestyle is not universal, even in this first of first world countries. That broadening of horizons is one of the main reasons I encourage people to try travel -- especially by bicycle.
I don't know if Dollar General has succeeded in driving "real" grocery stores out of large swathes of N.C. I'm guessing that if you went a few miles out of your way to larger towns and even small cities, you could find Food Lion, Piggly-Wiggly, maybe even Winn Dixie. Those are more likely (but still not certain) to have fresh produce.
A surprising number of convenience stores will have some of the less perishable fruits, such as bananas, apples, or oranges, located near the checkout. They're overpriced, but that's what you'll have to pay to support the low turnover.
As far as nutrition goes, my ancestors managed to survive on preserved food through the winters. I'd guess you'll have a fair selection of canned fruits and vegetables, even in small towns. You might want to supplement those with a multi-vitamin, although I doubt you'll suffer severe vitamin deficiency on a tour of less than 5-6 months.
I remember walking into a restaurant somewhere in Kansas and exclaiming, "They have a REAL salad bar!" After the western half of Missouri on the TransAm, that was a sight for sore eyes and a feast for a jaded palate.
