ccoker: Your barrel may actually be made by Marlin. I don't think Winchester actually made any of the 444 caliber barrels, I think Marlin made them. My Black Shadow, now Traditional Rifle, wears a marlin Microgroove barrel that wouldn't shoot fer snot until I 'Beartooth' lapped it.
The Timber Carbines came with a couple of different barrel twists, with one being a 1:20 and the other a 1:12, I believe. Some may also wear a microgroove barrel.
If you have read Marshall Stanton's writings on the 444 you will know that you can easily turn that Carbine into a tack driver with very heavy for the bore (300-350 gn) cast slugs. The 265-270 gn weight slug in the 43 bore is an excellent choice, but so are the heavier weight bullets. You may find that mid-range loads with heavy slugs are easier on you and more effective on your choice of game as faster moving lighter weight slugs.
If your intended game is smaller hog and whitetail within 100 yds the Timber Carbine would be my choice. Good luck. Mikey.