I feel hesitant to voice an opinion since all I hunt for is whitetail deer and if a black bear would come by, but I got a 35 Whelen just because I wanted one not because I needed it. Of the two calibers in question, I would find the 35 Whelen more in line with my thinking as it has manageable...
If it were me, I'd pull them and start over. If using an ammo case that is too tall for your shells, try putting folded paper towels or some soft material on top the shells, enough so that the lid closes. As for loose primer pockets, when I find them after I have seated a primer, I just write...
The throat in your Remington 700 sounds familiar as I have couple with long throats as well. What is the overall cartridge length with the Nosler BT? Curious how much difference between col and magazine box length. Sometimes rifles like a lot of bullet jump, sometimes not. Depends on the rifle...
My son uses 30-06 brass necked up to 338-06. I'd recommend annealing before necking up. Will help with longer case life in regard to the necks. Good idea to try to keep a little better track of your cases as to number of firings. Let us know how it works out for you.