I almost always carry at least 2 liters of water when I am hiking in this part of Arizona. Along the Arizona Trail, two liters should get you to the next reliable water unless you are hiking in the hot part of the year; not a concern right now. With two liters, you might get there thirsty but you will get there.

In my experience, icy trails are the major winter concern. Some parts of the trail will traverse steep, icy, north-facing slopes; snowshoes aren't much help in these places but traction devices such as Yak Tracks or instep crampons are really comforting. I doubt that you will encounter too much unconsolidated snow that is more than shin deep, walking that stuff is not much more work than is driving a pair of snowshoes IMO. The only place where I think snowshoes might help would be the stretch between Manning Camp and Italy Spring, in the Rincons, and in the Santa Catilina Mts. around Summerhaven.

Added later.
I don't think an ice axe would be much help. You shouldn't be in places where you need to cut steps or do self arrests. I used to carry an ice axe and unless I was actually climbing on steep ice and snow I used it mainly for balance; hiking poles will serve the same purpose. Hiking poles and traction devices would be the better bet.