Although I don’t have the same car as you I had exactly the same symptoms as you describe and hunted everywhere to find an answer. In the end it wasn’t until the problem developed a little further that I finally realised (with the help of a scanner) what the problem actually was.
Now, this may not necessarily translate into the same problem for your vehicle but I understand the E83 and E53 share the same components in this area.
What was causing my jerkiness was the fact that the transfer case clutch was slipping, which was caused by a worn gear on the XDrive servo motor.
BMW only sells the complete servo motor, but you can replace just the gear inside for a fraction of the cost and it’s a fairly simple job.
Hopefully, this is the same issue with your vehicle and that you can also get it rectified quickly and cheaply.
From what I’ve read, this is a reasonably common problem and none of the various symptoms people have written about with this fault were similar to what I experienced.