While you can't let baby plants dry too much, thyme likes it drier and it certainly sounds as though you are watering entirely too much. Just a guess here, but you can probably water half as much and have greater success.

You may also to water from below. Set the pot in a saucer or whatever to catch the water and fill that rather than watering from above. Unless the soil is bone-dry, the soil will wick up the water. You won't have "seedling abuse" that way.

FataMorgana

Potting soil, is NOT for starting seeds. It can contain all kinds of nasty stuff that seeds and young seedlings cannot fend off. Damping off, as mentioned in another thread is the most common of issues. Try a seed starting medium that is soilless and sterile. The seeds can be planted just an 1/8 inch deep. ALl them plenty of light and water lightly to avoid soak. Seed starting medium is light, airy, and will not clump up. These characteristics are very important when startin from seeds. I started thyme last year spring and by June, it was planted outside. Today is 2-3 times bigger than last year, and its all flowering right now. Using what I recommend will give you much healthier seedlines. I also start in trays with individual 2x2 inch pots and a clear plastc dome cover for the first few weeks, until they leaves touch the clear plstic dome. Then its removed and the indoor plant growing lights are on it from 7AM to 11PM.

Didn't catch the potting soil bit, ditto on what ksrogers said. Water comments still apply too.

If you want to, try starting a few seeds outside in the real live dirt too. Someone gave me packets of "herb seeds" for a gift a few years back and thyme was included. I wasn't keen on it since I had always had mediocre luck growing thyme before that so I just planted it in a little spot in the vegetable bed. Yeah...that's what I seem to do when I'm not interested in it enough to do the whole indoor seed starting bit. Well, low-and-behold, it sprouted. It grew. And a couple years later, it is the most vigorous and healthy thyme I've ever grown. I probably didn't have as high a germination rate as indoor starts but I still have 6 or so plants remaining from what I started from that packet about 3 years ago.

FataMorgana