Golf 7 Variant 1.6 TDI?

Hi, since the beginning of last winter I've had the problem that my Golf 7 1.6 TDI with automatic transmission always has a very rough cold start when it's cold and the outside temperature is below 15-20 degrees.
To be more precise, the engine shakes quite a bit when starting and then "wobbles" or "trembles" at speeds between 700 and 900 rpm. This subsides after about 15 seconds and then runs normally at idle, just like a new engine.
When starting off, it occasionally sputters, but that's not always the case. When accelerating slowly, it doesn't really respond to the gas. When accelerating from 30 to 50, the engine also knocks quite loudly. After 15-20 minutes, the engine warms up, and the problem disappears completely.
I went to the dealership in December for an oil change and mentioned the problem. They didn't find anything, just advised me to fill up with more expensive gas. I did that, and it got slightly better, but shortly afterward, it worsened again.
It was inspected last week, and I brought it up again. They said there's a sensor that can be replaced. I did it, but I haven't yet seen if it helped, as it's been quite warm in the mornings the last few days.
However, the nailing has been there constantly for the last few days.

Maybe someone has experienced something like this before or heard what the problem is. It doesn't feel good, and since I'm planning to sell it soon, I'd like to fix the problem.