I am not, by any stretch of the imagination, an expert on this particular issue, but my suggestion would be to move the thyme into a pot filled with potting soil somewhere outside in the shade. Make sure you're watering it regularly. After some period of time, perhaps a week, perhaps two or three, move the pot to the area where you intend to plant the thyme; again, water it regularly. Finally, assuming it's doing well there, transplant the thyme into the ground.

As mentioned above, it needs to toughen up outside. Low and slow...bright shade and kept damp, not too wet. It tends to thrive in rocky clay dead soil with good drainage once established. I have about a 1/2 acre area now without much effort. (zone 5a) frozen snow covered winters, hot summers(cooler nights)
If you have grass at this time, it may struggle against it. Compete, and not thrive.
Once you give it a permanent home spot, an edge area that won't be walked or trampled,
loosen up the soil and shake out all grass/weeds. Give it about a foot all around of space. It does like some sun.
No special soil or fertilizer needed. Once it toughens up, you can dance on it.

I let mine flower in August, then give it a mowing in october-ish to spread the seed all over.

Did you start that from seed? If so, you did great. You've got the touch. You might want to start a bit larger cell tray to give you more coverage. I have one that is about a foot square with i think 24 growing cells that is also self watering. I use them for small seed, slow germinators.
To really make a dent and establish an area of thyme, you'll eventually want a few dozen plants that size to give you a sizable and noticeable area of your lawn.

I changed my plans slightly so I realized I don't need a ton of thyme, just more than I have now. I'm trying to cover a 1'x~3-4' area in front of a small bush. It's an area I've been edging in, and have been weeding for quite awhile, so the soil is very loose, no weeds, but very clay-y, and doesn't have the best drainage.

The first batch I did split up into separate pots and I've been hardening off. They've been outside for a couple days straight, and they're doing better out there than they have inside! Most of the pots are showing decent growth for just a week, so now I'm going to try and slowly get them used to the direct sunlight they'll get all day.

I've also got another pot of thyme started (no germination yet), and I've spread some seeds directly in the spot, again, no germination.