If you are removing the full sheets of drywall you will have considerably less dust and of course your new ceilings will be like new. If asbestos is present, the demolition of whole sheets does not remove the risk, but reduces it. The peace of mind taking a $35 sample may be worth while. My experience is that it is unlikely you have any asbestos, but it is not outside the realm of possibility.
Jennifer, I just don't know of any way to remove the texture on your drywall short of replacing the ceiling or skim coating a smooth surface (difficult). You have the option of installing new drywall over the existing, or the preferred option of removing the old and installing new. In any event, the project will be messy and inconvenient. All the best. Hope you will keep us advised of your progress.