I can certainly imagine the effect you are feeling. I don't think it is frozen. The LSD has clutches that run in oil. Even at warm temperatures you will feel some significant friction - jack it up, put in some gear, and try to turn one wheel. You will feel a lot of friction when the other wheel turns the opposite direction. An open diff will do this freely.
When it is very cold, the oil in the diff will get a lot stiffer and the friction will go up a lot. At the same time, the tires are cold and will have less traction. And the road surface will have less traction due to ice (even a small amount lowers traction dramatically).
The result of all these condtions when you drive is you will feel like the diff is locked because of the high friction and low traction. Drive really carefully until the friction in the diff warms up the oil a bit.
