You're right. My bad.The power supply pins are correct on those 2 ICs (They have numbers against them.) I think it has just been drawn with the inverting and non inverting inputs swapped over to make the connections look neater on the schematic.
I have checked the power supply output with a multimeter and it is DC 12 volt. I should have said a DC battery eliminator.Just a thought, are you sure your power supply output is DC and not AC? In post 5 you mention a DC transformer. Transformers are AC. I would double check that as AC would most likely cause the "magic smoke."