Should I go see a doctor if I believe I dislocated shoulder? I think it's back in place already.

Subluxation. True dislocation usually can't be selff located unless you are a recurrent dislocator partial dislocations called subluxations if it dosent hurt and you have full motion no need to go to doctor if it is first tome see a orthopedist to avoid recurrent dislocations.
Yes. You need to make sure that the dislocation was not associated with a broken bone or a rotator cuff injury. A shoulder dislocation in any age group can be associated with a fracture. But dislocations in patients older than 45 are commonly associated with a cuff tear. Don't worry, the vast majority of patients with a first time dislocation don't need more than a quick check & maybe some pt.