Spring in the Smoky Mountains

Thanks for initial replies, great info. We are ok with pulling the trailer along narrow roads, was just concerned if there were some low bridges or tunnels that would limit the trailer. We are planning on going in late spring, mid to end of May so hopefully winter will be long gone by then with lots of spring flowers. Are there lots of RV's of various types travelling along the parkway? I had read that there was camping along the parkway, campgrounds without services. How do these work, first come first served? If so would they fill up quickly in May?
We were thinking it would be nice to spend the first part of our week travelling the parkway and in the national park, then spend the last few days in the more touristy area around Pigeon Forge & Gatlinburg.