Hi welcome to just answer.
From my experience, the speed of the player is controlled by motor and the weight of the stylus arm. From your description, it looks like the arm weight needs adjustments as it is pushing the record down too hard. The screw you described is the one that needs adjustments. Start at the center and reduce weight slowly until you feel that the weight is about to be off the track. Please use a damaged record to perform this as it can ruin good records.
Let me know the results