Just what time of year? Spring is March-April-May towards the south, April-May-June in the North.
I've been in the southern Appalachians during the traditional "Spring Break" mid-March, one trip was nice with a tendency toward chilly, the next year was cold and wet, miserable for the whole week. A year later, from same area and south to Great Smoky Mountains in early may, it started out chilly and drizzly, turned sunny and warm durn day, clear and chilly at night.
Several trips to Nashville March through June (while my daughter was living there 2007-2013) I saw the same variability, but late April was when it was starting to be consistently warm. In 2010, we were there for the 1000 year flood at the beginning of May, not only got caught out in the rain, but got trapped there for five days because we to refuge in a place where there was no way out.
I say plan to take this trip, but watch weather forecasts and be flexible, ready to cancel or change some part of it for weather.
Most of these trips were pre-RV, tent camping, so no recommendations for RV parks. A couple Nasville trips were RV, we used the KOA on Music Valley Drive, liked it. There are other parks along the drive as well. KOA got flooded in 2010, but that's not where we were that trip.
Tom Test
Itasca Spirit 29B