BMW E47 xenon headlights shine on the ground?

Hello, I'm driving my BMW e47 (1 Series) 118 d for the first winter. (I don't know if that's relevant to the question.)

My xenon headlights have somehow developed a life of their own.

At first they align correctly, then at some point they shine down on the ground.

Then they shine again for a while, then back to the ground and often they go up and down again the whole time.
I've tried everything. If I drive a bit faster on the motorway, they are suddenly aligned correctly. If I then brake, they always go down again, then sometimes even up bit by bit.

If the sensor was broken, they would have to be facing down all the time, right?

I also have the feeling it's only been happening since the cold temperatures arrived. Could the cold have an impact on it?