Crystallized honey is more than normal after winter. In cold climate almost all honey crystallizes in cold hive and feed sugar remains liquid.
So, it is normal that old honey and winter sugar exist in the hive before next yield. It it does not exist, the hive has starved.
In douple brood system job is easy when you turn over first and second brood box. Bees clean the upper parts of frames and make there brood.
It is good if they have a palce where to move the stuff, and that is whyn I make turn over when I add the first super.
It is good to move old store frames in the middle that when bees use them for brooding, they will clean them too.
Often there are too much winter food in spring, and it is better take off that queen has laying space.
You may use winter food easily to swarms. 3 food frames and the rest foundations.
If you have 1 or 2 winter food frames, you move then in summer between brood frames.