your shoulder dropping on one side is a sign that you have scoliosis which is a sideways curve of the spine, if its mild then your doctor probably wont notice it its usually dismissed as not being a problem unless the curves are severe, but it seems almost everyone that has scoliosis has some pain.
Manual therapy done by a chiropractor, osteopath, or a physiotherapist, is one way to try and correct the scoliosis, but if you choose therapy make sure you find someone good as there is the potential risk that your condition will be made worse.
Also you may have other issues an mri would help you find out if you have other issues such as ruptured disks or compressed nerves.
Good luck BumbleinGrey