If you've got workers laying drone brood, you're missing the queen entirely. Since April and May are the exact right times of year for hives to build swarm cells (queen eggs), ask anyone who has healthy organic hives if you can have one of the swarm cells. Cut the queen cell apart from the comb, leaving 2 inches in a strip attached. Then press the wax right onto a bar (you might want to warm it in the sun first so it's pliable and sticks to the bar. Then place the bar with the queen cell in your hive. The bees will take care of her until she hatches.
Soon as a queen cell is in there, the whole hive will orient to the queen. If you put here in there already hatched, they might turn on her, but if you put her in there unhatched, they will care for her and be prepared for her when she hatches, just like she was hive-raised. If she does hatch out ahead of time, you'd have to put her in a cage and let everyone get acclimated first. Unhatched you'll be okay.