Have you checked your pH? Thyme prefers a more alkaline soil.

How about watering? It should be allowed to dry out, then it should be deeply watered.

Too much fertilizer can also kill it - it is more likely over than under fertilized where that is a problem.

What is the condition of the plant when you pull it up? What does the root ball look like? Have roots spread out into the surrounding soil or can you see most of the original pot or root ball pretty much intact?

My pH is high. My "soil" does not retain water - it's mostly sand. I never fertilize my herbs.
Tomorrow I'll pull up the dead plant and look at the roots.

Take a trowel and try to dig it out - if the roots never escaped the pot, it'll be more obvious than if you pull it out and leave the evidence behind.

If your soil is THAT sandy, possibly it hasn't got enough water.

Thyme likes a pH of about 7, or 5.5 to 7. If pH is higher than 7 it definitely needs to come down.

What does it do as it sickens and dies?

There isn't much left of the roots, but what there is, did not grow in the shape of the pot.
The thyme gets drip irrigation twice a week and rain, mostly June - October.
The leaves fall off and then the branches dry out, starting with the outer-most ones. Sounds like not enough water, but now I think it might be too much. I noticed that it's been growing where it might get rain run-off from the deck. Would that kill it?

Oh yes....too much water kills more plants than not enough.

Thank you. I'll have to find a different spot for the replacement plant. Not an easy thing to do because I have a small yard where most places are shady. How much space does a healthy full-grown thyme plant require?

yeh with the thyme I'm growing, I realized it does better leaving it to almost dry then water, vs watering like every other day or day like I do other plants. I started noticing deadish/woody parts becoming more apparent, but that stopped once I stopped watering so much. Mines in a small pot with potting soil from nursery, think the pot is 8".

Another thing to consider - if you've been getting your plants from the same source over and over - try a different nursery. Maybe they just don't do thyme well.