Yes, many wild animals do, sheep have been bred for meat and wool for millennia, but the moulting gene is no doubt still in the DNA and in some individuals it appears to get switched on (expressed).
Interesting what breeding had achieved over time.
Those galls look like cherry galls, relatively firm if you try to squeeze them, oak apples are basically white but tinged green and red, they are soft like a marsh mallow, they are mutated stem buds so don't appear on underside of leaves. Curious you are not finding them with ease when there are so many here.
dw i'n gobeithio casglu y mis nesaf ychydig afal derw - I'm hoping to collect a few oak apples next month, (to see if i can get an image of the insect that causes them as they emerge from the galls).
Maggots could appear in a flesh wound, but we usually found them, either around their rear if not clean (spring grass can have an effect like too much curry)
Or from their feet (foot rot), when the sheep lay down you will see their front feet tucked under their body and maggots from their feet will migrate up the wool.
Keep the photo's coming.