I put together my setups before most of these ice fishing fly reel came out. I palm my reel and use a pencil grip on the rod, so I can't use many of those special design reels like the 621, black betty, etc. I use only mono, fluorocarbon is a bit too stiff and unmanageable in the cold for my liking. I am not sure if you want to use your fly line for ice fishing. I would strip of the fly line, leave the backing, maybe put on more backing to enlarge the arbor for more line recovery.
As for swivels, you could try to use a micro swivel from fishusa. I am not sure if that will cure the problem. Another way to stop spinning temporarily is to have a limber rod, or better yet one with spring bobber. When you shake the jig continuously ( very small shakes ), the jig will stop spinning. Spring bobber will dampen the movement of the jig, but still able to stop the spinning. Remember once you stop shaking, the jig will start to spin.
Fly reel is not for everyone, some will love it and replace all their setups. Some like my buddies will never able to get use to it.