It's hard to catch an egg eater. Took us months. We set up a deer cam and that gave us suspicions, but it wasn't confirmed until we installed rollaway nest boxes to stop it, and then we caught the bad hen going over to eat a couple of eggs that other hens had laid on the ground (in protest of the rollaway boxes). She went to Camp Amana, and we have had no more egg eating.
However, our girls have dramatically dropped production, too. It may be perfectly normal. Young hens will often lay well through their first winter, but just as soon as the days shorten the following year, they prepare to go into their first big molt. I think, based on experience, that their reproductive tract slows down before they start molting feathers (both happen during molting). This happened to us same time last year, and just started happening now. We went from 14 eggs a day to 4 a day in a matter of days.
We used artificial light last year starting in early August, and have already started it this year. It takes a couple/three weeks for the egg production to start to get back to near normal, though in fall/winter you may not get as many as in late spring/early summer even with light. We put a fluorescent light on a timer, and it is best to add light in the morning than in the evening. We have ours set to come on at 5:30 am (when it is dark). It's already helping. We had 7 eggs yesterday, and it hasn't even been on a week yet. The optimum is said to be 16 hours of light/day, but even ensuring that they get 14-15 helps. As soon as light drops below 14 hours a day, there goes your egg production.
This is one of the reason people who are most concerned with egg production get replacement chicks every spring, so they will lay through the first fall/winter without additional light. It's the mature hens that need it, if you want to maintain your egg production. There are only two bad aspects to adding artificial light and stopping the molt:
(1) If they look scraggly, they'll stay scraggly. We had to put chicken saddles on some of our low-ranking hens to help keep them warm last winter.
(2) Chickens, like every other animal, have a maximum number of potential eggs. If you keep them laying year round, for multiple years, they will stop laying sooner than if you didn't use artificial light--but you'll get the same amount of eggs out of them. Hens' productivity drops about 20% per year under normal circumstances.
Just some thoughts!