Sounds like a module as generally the pickup will not be effected by heat as quickly. You can take the module to be tested, if you do make sure they heat it up good by testing it several times. If you do replace it make sure you put the grease on the new module and be sure it is tight.
Please dont buy an Autozone or other replacement piece of crap. Look for someone who stocks Delco and get the right one. The OEM modules are far superior to most of the aftermarket ones.