How should I remove the texture from the ceiling *without* tearing it out?
I have the same thing.  We used two methods:  1st we used a sander to lower the high points of the texture,  this is something you want to use as last resort.  After the sanding was finished I did a skim coat, then had to sand again to smooth the skim coat.  Super labor intensive.
The next room I just put up 3/8 drywall, took half the time.  It seems more practical to just give up a little space and hang drywall.