Dying at idle?!?

Yes, don't throw parts at it hoping to get lucky. There are casinos over in Vegas if you want to gamble.

The suggestions regarding using a meter are spot-on. Lots of people have problems solving electrical issues. You have to approach electricity as if it is a road with cars on it.

If everything is good, the cars travel along smoothly and reach their destination. But, if there's a wreck, or roadblock or something else, what happens? The traffic stops, backs up, doesn't reach the destination.

It's the same with electrical issues. You have 12 volts at the battery and you want it to reach it's destination, which is the starter. So, what do you do? You don't install a new starter and hope the electricity gets there.

You start out at the beginning(the battery) and check all the junctions along the route. You inspect your route and insure traffic is moving smoothly. If you see a roadblock, then you know something at that point is amiss.

An electrical meter is a terribly underutilized tool. Not only will it measure voltage, but it can measure resistance, so it makes it dead simple to check a wire for continuity. You can go all the way down the route until you reach a point where it's obvious "hey, I no longer have 12 volts here" or "that side of that connection is live and that side is dead".

Good luck.