I haven't done much research on Stalingrad, but my undergrad capstone was on East Front logistics which invariably dabbled in the Battle of Stalingrad when supply lines were at their longest.

Antony Beevor wrote a great popular history about the Battle of Stalingrad. If this subject interests you, I'd definitely recommend it. Dan Carlin did a few episodes of his Hardcore History podcast called "Ghosts of the Ostfront," which I've listened to a few times while driving. What I offer here is primarily sourced from Beevor's work, but also other books and articles I've read over the years.

Food was in short supply since the encirclement began, however the shortage of ammunition was of far greater concern. Since all supplies had to be flown in, artillery pieces were rationed to a small number of shells a day which left the infantry without much support. The lack of fuel and spare parts meant that tanks became fixed fighting positions rather than a maneuver force.

Amazingly, despite the shortages morale was initially pretty high. You have to keep in mind that the German military was hitherto victorious in every campaign and had defeated some incredible odds. It only took about six weeks to knock France out of the war, something that Germany was unable to do in four years during the last war. Also, the Germans held the Soviets in very low regard after so dealing so many crushing blows against the Red Army.

Earlier in 1942, the Soviets managed to encircle a German force in the Demyansk Pocket. It was during this battle that the Germans were able to successfully implement the tactic of an airbridge. The successful supply and subsequent relief of the Demyansk Pocket was something at the forefront of the officers during the opening phases of the Stalingrad encirclement. The Luftwaffe eventually demonstrated an inability to deliver an adequate amount of supplies, which combined with the ever increasing distance to friendly lines led to despair in the city.

Beevor recounts some of what you stated in that some Germans tried to force their way onto planes or even cling onto the sides. The German military police, the Feldgendarmerie, were forced to devote a significant amount of manpower to guarding the airfield in order to ensure that only authorized personnel boarded the aircraft. The fact that the German Soldiers were so eager to desert is hardly surprising, as they were severely malnourished, not properly equipped or supplied, and faced almost certain death at the hands of the Red Army.