2.7A current with 10 turns is already ~2x the switching field for this reed switch. That is fine. Of course it depends somewhat on how you wind the turns, position and spacing. Also it is important to note, that if you use normally closed contact pair, such a circuit is more or less "fail proof" because if magnetic field is too small you will get indicator light, same as headlight failure. Anyway, you can use lab power supply to "calibrate" the number of turs for something like half (1.35A) of the current if you really wish. But 10 or 12 looks correct.
I'm a little confused about your wire getting hot, because for eg. 24 gauge, 10cm length copper wire gives me 0.0084 Ohms, and 0.06W of power dissipation at 2.7A, which should not be felt much at all. And for 22 gauge, 10cm, 0.0053 Ohms and 0.04W at 2.7A. There is not very big difference, both wire dissipation is very small. Maybe your 24 gauge is bad, or not 24. Is it both solid core and made from copper?
Any gauge is ok, which fits nicely in one layer for the number of turns required and has small power dissipation (less than 0.1W or so). I'm not sure about optimal spacing, I wind it directly on glass, or use thin electrical tape layer to stop sliping. Hope it helps.