We grew Thyme in barely zone 4 Montana when I lived there. I found that the young plants had a hard time making it through their first winter, and each year a much higher percentage survived. They get tougher as they age.
Also, some varieties are tougher than others.
Best thing to do is leave those dead looking twigs. Some will be dead all the way down and some will still have good lower buds, so wait until late June to snip out the dead parts with garden scissors.
RaeAnn had another way to do the Thyme which her tougher soul could do, but not me. Come October she "haircut" them down to the nubs, like an inch or two. She got a lower survival percentage, but also got prettier spring plants of the survivors.
Thyme are a highly variable group of species! Some are very low growing, perfect for mass plantings between stone steps. Others make miniature bushes, like a small Potentilla! Different flavors and scents, some strong, some hardly so at all. I like that Balsam.
My suggestion is, wait until late June before snipping out the dead wood. Decide how you want your plant to be, it's character. Fuzzy small growth or spiky with sparse heavy twigs. Keep her going with low nitrogen, good P and K type slow growth, and she should be stronger next winter. Don't let her tire herself with too many flowers. Pick some of them with a pinch.
