Crows thrive around farms and farm buildings as well as in their traditional roost woods. If you are going to keep numbers down to a level the farmer can live with you will need to shoot them in the woods and the farm yard.
Crows are very smart birds, no really smart! You need good field craft to shoot them in the trees where they have to advantage of height to see anything within shooting range. Young birds newly fledged may be shootable for a few weeks after leaving their nests but the get wise very quickly.
Decoying can work well but you need to ring the changes in your set up as they soon realise where the threat is coming from. Shoot different locations each time, mix up your decoys methods too. Flocked crow decoys on hedges, on feed clamps or out in the stublle work well. Crows usually come to decoys because they know they are not their regular crow friends, they come to drive them out. Some shooters have great success putting out broken nests with a couple of real eggs in. Try a small stuffed fox cub too, crows hate them and soon dive in on them.
Remeber not to send you dog for wounded crows, they are very likely to injure you dog too. If in doubt give a crow a second barrel on the way down to make sure it is humanely dispatched.by