Do these titles as used in Matthew 28:19 mean that there are three separate and distinct persons in the Godhead?

No. The Father, Son and Holy Ghost, actually identify the manner by which God’s authority (See: In the name of) has and will be given to us. It came from the Father (God, Mark 9:7), and given to us from the Son (Jesus, John 12:49-50), and will continue through the Holy Ghost (the Spirit of God).  We are led by God’s Spirit (John 14:26, Romans 8:9, 14, Gal 5:18, 25, ), which is given to obedient men and women (Acts 5:32) as it was given to Christ (Acts 10:38, John 14:10) while we await the return of Christ.