As of a 3 or 4 days ago you didn't even need snowshoes in the Adirondacks, so you certainly won't need them for the Catskills. I'd take a set of microspikes just to be safe, but you probably won't even need them. And I've never needed an ice axe in the Catskills.
Just remember that it is hunting season, so some really visible clothing is more important than anything else!