See if you can find at least one of the nails or screws that were used to put the ceiling or walls up. Are they magnetic? If so, a strong magnet run slowly across parts of the ceiling should reveal where the drywall was attached to the ceiling joists. You can then use a red marker or similar to mark the spot. You then find a few more to confirm the direction and location of the joists, and try to determine if they are 12", 16", or 24" apart.
However, covering the asbestos is not going to make the problem go away. If you ever plan on selling the home in the future, this is something that you must disclose to the real estate agent and any potential buyer. You also have the issue of screwing new drywall into the asbestos popcorn material, or having to cut holes for pot lights, a ceiling fan, etc. which will disturb the material and put fibers into the air.
While it's not cheap, doing a proper abatement and removing the material is the better approach than hiding it, unless you plan to not disturb it at all.