I use a braid on my tip ups, but tie on a piece of florocarbon.
I don't use a swivel, because most of the line twist is going to happen when you have the fish on.
After catching the fish, you can pull any twist out of the line when you wind it back on the tip up.
Stick with Florocarbon for leader material. it does make a difference.
Also, what size of hooks are you using ?
for inland trout, I use #16 to #18 treble hooks, for larger trout on the great lakes, # 8 to #12 work better.
Hope this helps,
Wiener