It would not be surprising if your rosemary suffered some freeze damage. Depending upon the cultivar (some are hardier than others) it will likely come back again in the spring. Don't cut anything back now, but wait until late winter/early spring to do so.

Our several years old rosemary shrubs have suffered the same kind of damage, for the first time.

I had a rosemary cutting from a friend's garden and got it to root in water. I planted it in a pot and the roots died. What did I do wrong? I used ordinary potting soil, was that the problem?

Rosemary can be cold/freez hardy , depending on cultivar. In my zone they over winter OK. I went to HD, while ago. They had some rosemary plants that had been cold stressed. The color wast not as green. But the plants looked alive. HD had cut the prices way down, to get rid of them. I bought one for about $2.50 and planted it in my garden. I think that the cold snap is over here and my investment might pay off.
The point is: your rosemary has been hit by cold. If the root is not badly damaged it may come back. You should have covered it with heavy blanket of fall leaves, to lessen the chances of loosing it. The winter is not over yet. Go ahead cover it, to keep it a bit warmer. The cold wind is worse than still cold air. That is why covering makes sense.

Rosemary tends to do much better with cold & dry as opposed to cold and WET. AS far as I know, the cultivar 'Arps' is best known for cold hardiness (think into zone 6b). It would also help if you grew it in a raised bed with soil having excellent drainage (add some coarse sand in the growing season). I wouldn't worry about some minor spotting. My rosemary bushes looks good this winter despite cold and wettness but the plants never got covered with snow so maybe that was the factor. Good luck!