Did the 3rd Fallschirmjager Division Combat with the 29th Infantry and 101st Airborne in 1944?

Jan 2019
United Staes
Exactly as the title. I am starting a reenactment group and I just wanted to be sure, as we are starting the 3rd Fallschirmjager's opposition groups, and those two groups are the supposed main ones to combat the 3rd Fallschirmjager in reenactments in a Normandy theme. I know that they all fought in the Battle of Normandy during D-Day, but each other? That's where I would love some help. :)