Julian Price and Crabtree Meadows are both on the BRP and are operated by the National Park Service. You will not miss them. Crabtree Meadows has a nice climb to do to get into the campground and it feels worse than it is because it's the last thing you have to do that day.
You may also be interested in Bear Den campground located at 324.8. This campground is a privately operated and nice campground accessible from the parkway, but there is no sign for it since advertising is not allowed on the parkway. This would be about 15 miles before Crabtree Meadows. They would have showers if you would prefer that.
Typically, climbs on the side roads going to the parkway are steeper than anything on the actual parkway. The parkway has long grades more so than steep ones, but they can be very long. You will face a difficult climb after Crabtree Meadows going south over Craggy Gardens.
When you get on the parkway at 270 you will in general be gaining elevation as you travel south until you pass Craggy Gardens. It is then an 18 mile downhill run into Asheville. Milepost 376 is Town Mountain Road and is the best route into downtown Asheville. If you stay on the parkway until MP 382 the parkway crosses US70, there is a grocery store there and several motels to the east within a mile.