Here is some quoted research from the late author Jentz:
http://www.feldgrau.net/forum/viewtopic.php?t=33216
According to Jentz, Die dt. Panzertruppe, Vol. II, Pz.Abt 127 on 10th of Nov., 1942, had the following units ready for service :
9 Pz II, 22 Pz. 38(t), 5 Pz III (kz), 10 Pz III (lg), 12 Pz III (75), 2 Pz IV (kz), 5 Pz IV (lg).
So for around the period of Nov. 1942, you have several choices for the panzer III version, as they had in their inventory both short (50mm) and long barrel (60mm), along with the support tank version N series (75mm).
Base colour at this time would still be gray, and you could add a white wash if doing a later in winter time frame. Crosses would be in typical application, just look at photos of other units for this time period. Turret number style, maybe some photos of 22nd panzer can give some ideas. Again, since no photos of the 27th exist, then that is also in your favour, as no one can disprove any choices you make with this subject.
regards,
Jack