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The most desirable kind would depend upon what you want to do with it. Do you want it as ground covering, for a kitchen herb garden...there are countless uses for thyme. 'Coccineas' has magenta flowers in early summer and is often used as a grass subsitute or tucked in between pavers.

Here's an article that may help you decide:

https://www.gardeningknowhow.com/edible/herbs/thyme/types-of-thyme-plants.htm

This is an infection, rather than going woody. Woody stems are a natural part of aging, and will occur, only, on older growth. This new growth that is effected by a disease. This, usually, happens with overwatering or soil that doesn't have a chance to dry out thoroughly down to about 2 or 3 inches between watering.

I would treat with a fungicide or change out the soil and treat (which will be more thorough).

Here are some articles that will help:

https://www.gardeningknowhow.com/edible/herbs/thyme

https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

It is likely your soil. Thyme requires poor soil, with very little nutrients, and dry conditions. Nutrients that are too concentrated, or soils that have too much nutrients can cause this.

Here is an article that will help you to care for the plant:

https://www.gardeningknowhow.com/edible/herbs/thyme/tips-for-growing-thyme-in-your-garden.htm

You will want to only plant the plant in a container that is only an inch to a few inches bigger than the rootball itself. This will help prevent overwatering, which is likely the reason that they are dying off. It is still possible to grow them in large containers but you must be careful and only water them when the top of the soil is completely dry down to about 4 or 5 inches or so. This can take quite a long time in a large pot with very little roots to take the water up.

This article will help you to grow thyme in container: https://www.gardeningknowhow.com/edible/herbs/thyme/growing-thyme-indoors.htm

Yes. I see now. My answer still stands. It is a root maggot, many of which cannot be distinguished from one another. The articles that were posted in the answer still apply, with one addition.

Your roots are likely going to rot, both, from the infestation and overwatering.

Likely, you will be left to start over, even after treatment. Along with the treatments from the other question, I would recommend a fungicide.

https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

Early spring is the best time for planting creeping thyme. Here's an article that will help: https://www.gardeningknowhow.com/edible/herbs/thyme/growing-creeping-thyme.htm