You need to identify the critter.
Take photos of the mounds.
Smooth them out and check at various times of day to determine exactly when the damage occurs.
And fill out your zone information so it shows when you post so we know what geographic area you are talking about.

A shaded area is not automatically a damp area - that's much more a function of drainage or soil compaction than it is shade. Most lawns are not at all happy under very shady conditions and certainly not if very moist or if drainage is bad.

Earthworms leave small 'dirt' mounds - essentially their castings. They tend to do this in spring or after wet weather and then stop. And you DO NOT want to get rid of earthworms. In addition to the value their castings add to the garden, they aerate the soil.

You don't indicate how large these soil mounds are but various other creatures create mounds as well - moles, gophers, armadillos, ants, etc. You will have to suss out the creature and then you will know how to control. But I'd first investigate ways of improving drainage and soil conditions in this area and lawn alternatives. You are never going to have a satisfactory result until these issues are fixed.