No, you don't have to,as long you don't sell the badges of vows and keep them.
All quests, including the Sky Quests reset in a NG+ and can be done again.
But also all your items, affinity ect. pp go over to the NG+ including everything that is in your inventory andall the stuff that is at the inn storaged.

So when you stuff all the badges into the inn storage and go over to NG+, all the badges are still there that you've collected and once you retake the Sky quests, you can basically quickly get some free Exp. and Rift Shards as you've not to go out searching for the Badge again.
Just take it out of the storage and basically the Quest should instantly update as being done.

Thats the basical theory that should work this way, unless Capcom has made for DLC Quests any unknown exception.