Ohm's law isn't the right law to use to prove or disprove transformers. It's like using the law gravity to prove water boils at 212º. Good law, but not really the right place to apply it.
I guess the best law to justify what transformers do is conservation of energy. Watts in = watts out.
But Ohms law does apply to each coil of a transformer, as long as you're taking your measurements on one complete loop, that is through only one coil at a time. Because the two coils (or however many there are) aren't directly attached but only interact by magnetic fields, ohms law doesn't work across a transformer, only through a continuous path such as through one of the coils of a transformer.
And ohms law says I=E/R, so current through a coil goes up as voltage on that coil goes up, and goes down as impedance on that coil goes up. Impedance goes up as the magnetic field (from a load on the secondary coil) fights the current in the coil you're testing. So ohm's law does apply, just not quite in the way you were thinking.
Man I wish I was better at explaining stuff.
