The Lufwaffe Felddivisons are something totally different from the Fallschirmjäger Regiments.
Germans used Fallschirmjägers as such until Crete in 1941 (with only 2 exceptions afterwards) but they used these troops in combat nevertheless.
First, in September 1941, the II. Fallschirmjäger Sturmregiment was send to Leningrad to fight as ground troops. Soon after the I and III followed. In summer 1942, there were already several Luftwaffe Regiments deployed in North, Center and Southern Russia and in Norway.
Then Hitler gave the order to the Luftwaffe in September 1942 to send 200.000 men to the Heer to compensate for the losses in the east. Goering made him instead agree on raising 20 Luftwaffe Felddivisionen, separate from the Heer. Until they fell under complete Heer commanding in November 1943, they even had several Luftwaffe Feldkorps (commanding divisions). Those Troops were equipped and supplied by the Heer (not artillery, they used their own 88 Guns). They even rejected Weapon Training Personell from the Heer and so the Luftwaffe trained their own Ground Troops.
Bad Training was one of the reason that the Soldiers from these Divisions were not much liked among the Wehrmacht Soldiers, as they were in general bad in combat when they arrived in their shiny uniforms and with new weapons on the Front, just to die in the first combat because of the bad training.
