Dolomites don't have enough snow, not even after the last layer. Moreover, this is more of a winter vacation than winter sports: the best cruising, food and surroundings you'll find in Euope, though.
I would suggest Monte Rosa, Italy (fly to Milan or Turin). Runs for everyone, great off-piste as well. I think @prickly can comment best on snow conditions out there, but I think they're great and will be in a month as well.
I would like to suggest Espace Killy as well (Val d'Isère and Tignes).
Austria is great right now, but is relatively low and has many south-facing slopes generally, that gets a lot of sun and heat as well. If you have to book so far in advance, I would go for Monte Rosa or Espace Killy. Chamonix is also an option, but is a different kind of skiing (more ski-mountaineering, or bus-hopping form little resort to resort).