My Bilstein rear shocks have the piston area (where the dust boot is) up top, but keep in mind I have Tundra rear shocks with an "eye" on the bottom and "stud" on top. No adapters here.
The LC coils are known to be pretty tall, and notoriously so when they are first installed, especially on our lighter T4R's. It took mine approx 3-4 weeks for them to actually settle to where they sit now and that was with me driving around with my sleeping platform, a few tools, full M/T spare tire in stock location, and 2 of my LC80 wheels with KM2's on top of the sleeping platform. I had to weigh it down for a while to get the settling I wanted out of the rear springs. What does this have to do with you??????
Regarding the seating of the coils: Do you have a adjustable panhard bar installed? Or a panhard drop bracket installed?
I am thinking that you just have the normal HUGE amount of lift accompanied with initially installing LC coils in the rear. Without the panhard drop or adjustable panhard bar, the axle at such lifted height is going to be quite a ways off to the side at static height. Think how the suspension articulates with our 5-link design and how the axle actually moves to the side as it follows the rotation of the panhard bar. So, maybe because your axle is ssoooooo far out of stock range with the new lift and stock panhard setup, it is causing the coils to want to sit crooked (not perpendicular to the axle) and not seat correctly in the coil buckets. This would also contribute to the "twist" in your rear shocks. I am thinking a panhard that is longer (an adjustable one) or the panhard drop bracket will bring the axle back to the center where it should be: pulling the axle back towards the driver's side. This would make the coils perpendicular to the axle once again (or close enough so that the coils will seat properly).
If you have a panhard drop or adjustable panhard already installed, then I am at a loss....