First week in January is likely to be busy/expensive as lots of folk come up to see in the New Year. You might be lucky to get a part-week once some have left.
Certainly no shortage of accommodation in the area generally. A small house in the Dalnabay/Silverglades area isn’t rustic or cute but will be warm, lean and relatively cheap.
Access is almost guaranteed as it’s on the A9.
Also look at Kingussie and Carrbridge. Not too far away but might save you a few bob.
Assuming we have snow then Cairngorm itself is fairly straightforward once you dodge the skiers. If the snow us low enough then there’s a decent sledging area at the Hayfield in Glenmore.
It’s difficult to think of somewhere else with complete an offering as Aviemore. Braemar/Ballater possibly, Fort William/Fort Augustus, Pitlochry at a push.