I just desperately don't want to hurt my shoulder(s) really bad
The more afraid you are the less you will commit and so the more likely you will be to stack yourself on the point of your shoulder and hurt yourself.
Chin on chest and go for it and your back will tack the impact, but its designed to do that so it will be fine.
For a slightly different take on this, zempo kaiten is a more advanced falling method. I do not have beginners attempting them for a while. If they have problems, I do not push the zempo kaiten but focus on the side and back falls so the student gets used to falling on their own via various drills, and being thrown eventually.
There are plenty of drills and exercises to do to prepare for them. No hurry. And as pointed out get help from your instructor.