I'll disagree with MerryKalia about Gatlinburg. You can pull a trailer through the area as the roads are nice without any tight corners. However, I would not WANT to pull through there from noon to midnight because of the bumper-bumper traffic and all the pedestrians.
For Smoky Bear and Jellystone campgrounds, take I40 exit 435. In about 1 mile stop for fuel (best price in eastern TN) and food at Walmart. This is the last decent place for food. Continue on down the road, right at stop sign, few hundred feet then left at next stop sign. Follow the signs for Gatlinburg. Smoky Bear will be on the right about a mile past the apple orchid. Directions work well for Jellystone too but not as far as Smoky Bear. I would not recommend pulling the trailer over the Foothills Parkway as it is just one long steep climb on both sides.
I personally would drop Myrtle Beach and give more time at the other stops. If the kids were older I would say yes, but such a young age the extra miles doesn't seem worth it.
You know your kids best, but when ours were younger, we found getting up early (5am) and traveling with stopping around lunch worked out best. They would tend to sleep for the first couple hours and then would be getting restless by the time we stopped. But the important thing is they had the afternoon to run around, go swimming, etc to get all tuckered out in time for bed. Plus campgrounds tend to like folks coming in at 1pm much better than 1am not to mention easier to negotiate them.
