Is it possible for an adult to "Dry Drown" from choking on a cup of water?

Theoretically. Anything which triggers a laryngospasm can cause "dry drowning". I believe there would have to be an underlying illness for that to occur however, most people have a reflex in the larynx to prevent the water from going into the lungs, but if this spasms (usually from prolonged or persistent exposure to noxious gasses or fluids) it could lead to death.
System Of A Drown. "Dry drown" means little or no water gets into the lungs which means the throat (glottis) goes into spasm and so no air gets in either, which means you die unless the spasms break before that happens. As someone is dying the spasms relax but by that time it may be too late, especially if underwater. This sounds unlikely in the case of "choking" out of water, unless something got stuck in there.