Kent J. Yes, I think that you have a queen, especially when you said that you saw one egg per cell. It is known that when workers lay eggs, the eggs are not situated right in the centre, as the queen does. Their abdomen is too short and cannot reach the bottom of the cell, hence the eggs on the side or misplaced.
It is hard to see the queen sometimes, I have had a couple of hives that when I checked I could not find the queen for love nor money, I saw eggs, (positioned properly in the cell), so I know that the queen was present at least around three days prior. Have a wonderful day, patience, great life. Cindi