The regular hiking season is definitely over (bus + hut), but this doesn't mean you can't hike there. You just can't hike in a crowd ;) and maybe with some snow. A little. Most likely not more than a pair of gaiters can't deal with. It would be most unusual if you would need snow shoes, after all the dolomites are not that snowy. Normally I fear that there is not enough snow when I go ski touring there in high winter ...
The big routes via the via ferrata at high altitude is most likely really over, but part of the charm of the dolomites is the space between the high peaks with trails that are technically not so demanding and offer nice views on the mountain scenary.
The trails are usually quite abundant and offer plenty possibilities for plan B variations in case the planned route turns out to be unpassable. Just plan your route accordingly.
The 'best'maps you can get are the Tabacco maps . They quality is not what most usually expect from a topographic map - but it is ok for finding your way home (usually) :)
In some areas of the dolomites are plenty of "bivaccos" as shelters. These are small emergency shelters that you might use if you plan an overnight trip. As they are usually in the higher places you need to check of course if the conditions allow getting there.
Here a few pics from the Sennnes-Fanes Group (northern dolomites) in late October 2002 after the first heavy snow fall.