If you're logy, you're slow and listless, not reacting quickly. Most coffee drinkers start the day off in a logy state, before they've had their first cup.
Hot, humid weather makes some people logy, and so does missing an hour or two of sleep. When your dog wakes from the anesthesia after having his teeth cleaned at the vet, he might be logy and groggy for a while before he can walk in a straight line. The adjective logy, coined in the US and primarily used in North America, is fairly uncommon today. The root may be the Dutch log, "heavy or dull."