I've seen the ignition switch do this, but I usually expect other problems associated with a bad ignition switch. I'd hate for you to replace it, and that not be the problem.
I'm sort of miffed by the starter relay having burnt looking pins too. All that relay should do is provide a way for the PATS and NS switch to prevent the car from starting. The inertia switch cuts off the fuel pump, so it's obviously working. The starter relay breaks the hot going to the solenoid (the real starter power relay) so that the car won't start either in gear or without the transponder signal from the RFID chip in the key.
Is there anywhere in your area that rebuilds starters? Look in the YP and see if you can find a local rebuilder, either that or ask at local shops. That's who uses those, and what keeps that business in business. Most of the time a local rebuilder will rebuild your starter or alternator cheaper than you can purchase one at a parts store. The work is typically more reliable too because returns are more damaging to small businesses. We've found that local alternator rebuilds are much more reliable.
An old man somewhere in the back of my mind is telling me that an engine that won't start hot has a problem with the starter solenoid.
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