Yesterday, we did a review of Tai Otoshi and Ogoshi. I think I finally understand the mechanics behind Tai Otoshi and it's pretty comfortable to do the throw now, although I tend to turn my torso further than most people in my class when I do it.
I also learned an advanced variant on Ogoshi where instead of pivoting and then squatting, you squat first when you apply the kuzushi and corkscrew your bottom around 180 degrees. This has the effect of accelerating the throw dramatically, resulting in a much more explosive technique. I love this variant and it's much easier for me to perform than the standard pivot and squat method.