If you road is free from deep snow then the touring bike is fine.
Tourers can handle light snow although it can accumulate between the wheel and fender. If you have to lose the fenders, you can fit clip-on ones with greater clearance.
To winterize your bike, clean and wax the frame, ensure all thread and metal-metal contacts are greased (or use anti-sieze). Wax your exposed cables and pack the enclosed cables with grease.
Pack your hubs with grease.
You may want to fit a winter tyre, either an aggressively treaded touring tyre or a studded one.
Non suspension MTB/commuter bikes are making a small comeback. These are ideal for winter commuting although they have very little advantage over your Fuji tourer.
Just keep riding and see how you cope.