It might be possible to save part of 6th Army, but it sounds like it would be a matter of a bloody, brutal rear guard - probably a running action and rear guard, and the pieces surviving either being the fastest (not necessarily best) or skeletal formations, and probably both.
And this is assuming things work on the Axis end, which is not a given in these conditions.
Add in Soviet competence and it sucks even worse, so I'm just looking at what the Germans have to work with, which is already too little.
Doom sounds a little harsh, but any possible fate for Sixth Army is not going to be a good outcome. Just "Total disaster" with OTL being not even the worst scenario (Imagine a costly relief effort plus OTL's losses) on one end to "unaffordable disaster" where men escape but not much in the way of equipment as the very best case scenario.