1. Coils don"t have "internal resistors". Coils designed for use in a 12 Volt system are wound with more wire than coils that need an external resistor.
2. Coils do not "shut down" when they warm up. If they do, they are defective.
3. 3.7 Ohms is more than adequate primary resistance IF your meter is accurate at these low resistances. What does the meter read (in Ohms mode) when you short the test leads together?