If you’re staying in Avoriaz the closest sheltered runs are down to the Secteur Linderets and onward down to the Ardent gondola.
The latter is a perfect place to park if you are staying in Les Gets or Morzine and have your own transport. Alternatively access the area via Super Morzine, Express Zore, and the telesiege Seraussaix. This will place you above Avoriaz and Linderets itself and offers up multiple options.
One of the best places when the clouds move in and the snow is falling is The Stash snowpark. Lined by the Lindarets trees and with multiple route options, you can easily spend all day in here without getting bored.
In terms of piste, the Combe a Floret and Les Tannes are fairly enclosed runs offering the shelter required and have multiple off piste cut-throughs for more advanced skiers and boarders. To repeat the various options use the Express Linderets to return you to the top. Further along the same mountainside the Express Prolays lift opens up more reds and blues again sporting multiple lines and cut-throughs off the pisted runs. The edges of the piste throw up many opportunities to get briefly into the trees and back to the piste without running the risks of hitting more demanding terrain further from the marked runs.
On the opposite side of the valley, the new express version of the Chaux Fleurie will whisk you up to the exposed Col du Bassachaux where you can avoid a whiteout day by dropping down the multiple lines under and near the lift line. Caution, some of these lead to some smallish cliff drops putting this off-piste out of reach for the less experienced who would be advised to stick to the Grand Plan blue and Chaux Fleurie red converging at the bottom of the aforementioned lift.
Don’t ignore the home run down to TC d’Ardent, the Parchets blue, although not a demanding run, it is extremely entertaining with playful areas on each side of the piste. In particularly bad conditions it’ll keep you entertained for multiple runs without having to freeze on an exposed chairlift in between.