My two picks out of Zurich would be Stuben and Davos/Klosters. Stuben is low on glitz, small, and very quiet, but it is linked into the St. Anton lift system and a short shuttle ride over to Zurs and Lech. It's less expensive than its glitzier neighbors. The Albona, directly above Stuben is one of the great lift served north facing slopes in the alps. Recommend Hotel Mondschein. There's a direct bus (Arlberg Express) from the Zurich airport which will drop you (almost) at your door. Much easier than going on the train. No luggage schlep.
Davos/Klosters is an easy train (2.5 hours) ride from Zurich. Snow sure, tons of variety so you won't get bored. Davos is not a charming alpine town. Klosters Platz is the choice for charming. Hotel Meierhof in Davos is a good choice on the higher end. Hotel Parsenn in Davos is a 3* with a great location opposite the Parsennbahn. Chesa Grishuna is Kloster Platz oozes charm (and it's Chalet annex) and has a great location near the Gotschnabahn. Hotel Silvapina in Klosters Dorf is more humble, traditional chalet style, more reasonably priced but not as centrally located.
A bit farther afield from Zurich is the Ober Engadine (St. Moritz, Celerina, Pontresina). St. Moritz invented glitz, but if you stay in an outlying town you can avoid the over the top stuff in St. Mortiz, enjoy some fabulous skiing, and cut your expenses somewhat. Zermatt is a longer train ride from Zurich as well. Wonderful place, tons of skiing, pricey.
Ditto on your timing but I'm guessing that that's something that you can't control. By going in March, you avoid some of the crowds and the hotels in some resorts dial back their rates a bit. The Arlberg, (Lech, Zurs, Stuben, St. Anton) is always busy. Engadine is the quietest on the slopes of the above places.
No bad choices. By all means go. If you're going to a Swiss resort, you can check your bags in NY directly to the train station at the resort. The hotel folks will go and pick them up for you and take them to your room. You can do this through Rail Europe at a reasonable cost. If you're flying on Swiss, you can do the same thing on the return by taking your bags to the train station the day before you leave and checking them through to NY.