Does your pot have drainage holes? I check the dryness of my pots by lifting the pot up. Does it feel light? It might need to be soaked in a pan of water so the soil gets fully wet.

Is the potting medium moist or dry? Turn the plant on its side and gently slip it out of the container to check on how it feels throughout the whole volume.

We can't tell if it's too wet or dry; only you can do that by inspecting.

Since only part of the clump is flopping at present there is probably a problem with the roots on that particular section. It is likely that what is in the pot is a cluster of seed grown plants rather than one single one. If you gently tug one of the floppy parts does it come away from the soil? Have the roots rotted?

Bottom line is that Thyme is really not a good candidate for growing indoors. Even your brightest room is not going to provide anything like the light of a Mediterranean hillside, which is where your plant really wants to be. It need to have masses of light and a very well draining medium to stand a chance inside.

Thyme needs lots of direct sunlight. If you want to grow it indoors, you'll need intense (bright) lighting very close to the plant.

I grow rosemary, thyme and some other herbs indoors in the winter. Here's a picture of one of my rosemary plants in the indoor hanging planter I have for it.

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Note that I have a very bright, very focused grow lamp about 12 inches above the plant, and this is actually hanging in a well lit kitchen, under a 4 tube fluorescent light that stays on all day. That is a 9 watt, 9 LED grow lamp, and it's as bright as a 90w halogen flood light. Both the room lighting and the grow lights are on a timer and run 12 hours per day minimum.

With any less light, my herbs barely grow, and when they do, they grow very 'stretchy' and weak. They need intense lighting, close up, to thrive and grow.

Exactly, lots more light is required than a normal indoor environment.