I have used Gallop to kill grass and moss. If I just dig the garden over leaving all the dead moss and grass will it make matters worse?
Do not go to the trouble of digging. Use sulphate of iron to kill the moss and re-sow the grass with No. 2 seed when it is dead. The seed can be scattered on the soil surface, even onto dead grass.
In future, do not mow as tightly and apply some lawn fertilizer each spring.