This is a common misconception about all popcorn ceilings. You will be fine. It is a very messy job. Good luck.
Asbestos was banned from residential texture in the late 70s although existing stock was allowed to be used up. Anything after the early 80s is presumed to be asbestos free. I wouldn't worry any.
Has the texture been painted? Painted popcorn is harder to remove but it can still be done.