Could you have a dislocated shoulder and full range of motion at the same time?

No: There are two main types of shoulder dislocations: anterior and posterior. Either type of dislocation is very painful and always associated with a limitation in arm movement. I have never seen a shoulder dislocation where there was still full range of arm motion.

Unlikely: When your shoulder dislocates the ball is out of the socket and the muscles don't work to their full effect making it virtualy impossible to have a full range of motion with a dislocated shoulder.