As lexma said, driving from Lyon to Albi via Carcassonne is a long trip. Also, it depends on where you start in Lyon; leaving on a weekday morning (like a Monday), you'll likely run into the usual commuter traffic. So a 6h estimate with no stops is extremely optimistic, even for a local used to this route.
Given these conditions, I don't see how you could visit Carcassonne in a single day and push on to Albi, especially while enjoying a cassoulet. The post-cassoulet food coma is much better suited for a nap than driving, especially after hours behind the wheel since early morning. :-) In my opinion, the wisest and most practical option is to stay overnight in Carcassonne.
From there, it comes down to history: as lexma pointed out, the medieval city was restored in the 19th century under the aesthetic direction of Viollet-le-Duc in a pseudo-medieval style with heavy Gothic influences, ending up looking more like a Disneyland castle than its original self. Take it or leave it.
The Cathar castles (even though they aren't actually Cathar) have remained in their original condition. Visiting them would take several days given how spread out they are across the region, and some require a bit of a hike since you can't drive right up to them.
Albi is a nice town, but I don't think it warrants more than a full day and an overnight stay. The city is lit up in December, and there's a video-mapping light show on the cathedral in the evenings, typically starting around the second weekend of December. The same goes for Castres, the region's other major city.
Albi and Castres in December 2025:
https://photos.app.goo.gl/rjobBoZ1G5pBM5KK6
If you stick to your itinerary, I can point out an easy and pleasant way to get from Carcassonne to Albi while bypassing Toulouse (along with a few good spots to stop along the way).
As for the Dordogne, don't make the mistake of hyper-focusing on Sarlat, especially coming from Albi or nearby. Sarlat is located at the far western edge of the Dordogne valley and isn't inherently more interesting than many other spots in the Dordogne that are much easier to reach from Albi. Just because Rick Steves only visited Sarlat doesn't mean it's the only place worth seeing. Sarlat deserves a half-day visit, but nothing more. Once again, I can show you a smart and interesting way to get there from Albi.
Keep in mind, though, that this will be in December. Don't expect to stroll around in the sun through charming villages with packed café terraces like you see in travel magazines. From a tourist perspective, quite a few things will be closed, rivers activities like kayaking & excursions are suspended until April (at least), it can be cold and rainy, and it gets dark by 6:00 PM. People aren't wandering around outside, for the most part, they head straight home after work to watch TV. Agricultural activity in these rural areas pauses for a few months.
I feel bad pointing out all these downsides, but I just want you to have a realistic idea of what to expect.