If the car won't turn over (or "crank") I'd think the RCM is the problem. Re-soldering usually fixes it.

But if you car has the original battery, replace it NOW, if it is NOT the problem, it's going to BE a problem. O'Reilly's had one that meet the specs for $89.

The Crank Sensor can set the light, but not always. If the light comes back on, go to Advance Auto and get them to read it (they do this free of charge).

In any event, you are describing a bad RCM.
Contact tight1 on this site, see about getting an exchange RCM from him!