Store gear at Newfound Gap

[HooKooDooKu]

Gattlingburg would be a good option and a easy hitch, however you can also resupply there also so no need to stash gear.

OP has a shuttle to Clingman's Dome from which he will start a section hike that (I'm guessing) will take him to Davenport Gap.
Since the shuttle will have to drive by NFG, they had hoped to be able to drop packs off at NFG, "day hike" from Clingman's Dome to NFG, then pick up packs and continue towards Davenport Gap.

But since that are arriving by shuttle, they will not have a vehicle at NFG and therefore will have no good place to store gear for the hike between CD and NFG.

[Starchild]

OP has a shuttle to Clingman's Dome from which he will start a section hike that (I'm guessing) will take him to Davenport Gap.
Since the shuttle will have to drive by NFG, they had hoped to be able to drop packs off at NFG, "day hike" from Clingman's Dome to NFG, then pick up packs and continue towards Davenport Gap.

But since that are arriving by shuttle, they will not have a vehicle at NFG and therefore will have no good place to store gear for the hike between CD and NFG.

Gotchya, not worth the trouble to figure that out then, it's a easy hike from Dome to NFG.

[Harrison Bergeron]

I don't get it. Clingmans Dome is SOUTH of Newfound Gap and you said you want to hike North. Why would you want to stash your gear behind you? Besides, that 7 miles to Newfound Gap is easy-peasy, all down hill, and practically follows the road. My kids even joined me for that bit. Are you confused or am I? Sounds like you really want to stash your gear at NFG, not Clingmans. But not necessary. You can drive almost to the top of Clingmans. Just leave your gear at the Clingmans parking lot with whoever drops you off, while you run up to the top for a picture. Then, when you get back to the parking lot, pick up your gear, say goodby to your ride, walk down the road about a hundred yards and then into the woods west for about twenty yards until you find the trail. Turn right, and hike on.

[HooKooDooKu]

I don't get it. ... Are you confused or am I?

I believe You are.
Did you read Post #41 above? It lays what OP is trying to accomplish.