Well, my gut instinct is that since that's not normal behavior for the start switch, that's the first part you want to test. Time to get out the ol' multimeter. You'll want the shop manual that has the circuit information for the switch, so you can test the correct continuities.
If you take apart the switch for any reason (or none), be careful because it's probably spring loaded. Some people recommend disassembly inside a bag to contain parts going "sprung".
