I've done the week in one resort and a week in the other. Getting the connecting transportation figured out is the hard part. Last year I spent a week in Pontresina then a week in Stuben. We were a group of six, so it was cost effective to hire a taxi to take us from Pontresina to Stuben. I priced a Swiss taxi and an Austrian and went with the Austrian. Could not have been easier. Took the bus (Arlberg Express) back to Zurich from Stuben. Essentially door to door service, hotel to airport. If you are a twosome, train and post bus are probably the best option pricewise, but that involves schlepping bags on and off trains, which some folks don't want to do. The combination of a week in the Engadine and a week in the Arlberg is a tough to beat one two. The Engadine doesn't get as much love as some areas because the glitz of St. Mortiz is a bit of a turn off. By staying in other nearby towns (Celerina, Samedan, Pontresina, Sils Maria) you cut down your lodging costs but still have access to amazing terrain that does not get that crowded. Stuben is the same play in the Arlberg. Very sleepy town, lodging costs lower that St Anton and Lech, nice hotels, and has a lift that links you into St Anton (not to mention accessing some of the best north facing lift accessed off piste in the Alps) and easy bus connection to Lech/Zurs/Warth.
There are a lot of great off the radar areas in the Alps, but they probably will not keep your interest for a week. That's why the mega resorts are so popular. You can try out new areas almost every day. I would agree that western US snow is better and more plentiful, but the infrastructure in Europe makes the off piste skiing much more attainable. You can ski off the back of most any lift in Europe and, if you know where you are going, you'll come out at a bus stop or train station that get's you back to town or to the next lift in a matter of minutes. I'd second the guide option. There's too much great skiing to be had to try to figure out things on your own. I've spent over 30 years touring and off piste skiing in the Alps, and I do not trust myself to find the best route, the best snow, and the safest route.
If you fly into Zurich, a week in Klosters/Davos and a week in the Arlberg is easily managed by train or bus and gives you a Swiss experience and an Austrian experience without a lot of time or cost getting from area a to area b. Feel free to PM me for resort info. I've been to most of the bigger places (not too much in France beyond PDS, Chamonix, Tarantaise) and many of the smaller ones so I have a good handle on geography, transportation options, costs, hotels and guides.