You can, in fact, buy studded tires that will fit your cross bike/MTB, and they would allow to you to continue the snow riding you have been doing, as well as provide you some traction on the ice. Beware, however - snow over ice will usually still result in a fall, even if you have studded tires due to the tire floating on the snow, and not gripping into the ice below.

I have been riding in the snow for years, typically on my MTB, with and without studded tires. This works in good conditions, but not all conditions. Fat bikes (yes, I have one, no it is not a regular MTB) have 4-5 inch tires that provide a much bigger platform for "floating"on the snow. These will let you ride in some conditions where a regular MTB wouldn't work. The race I did this weekend was a great example - much of the single track would have been fine on a regular MTB, but the snowmobile trail connecting the course, and some of the field sections would not have been rideable without a fat bike - the snow was simply too soft (and it would have been a LONG uphill walk!). Think of riding in sand for miles... The fat bike lets you do this where a regular MTB would not.

Whatever gets you outside and riding is good in my book! FWIW - I hear that Schwalbe makes a pretty decent studded tire for the cross bikes.

SheFly