Two possibilities: 1) the infectious agent was not bacterial but was viral; or 2) the infectious agent was bacterial but resistant to the Z-pack. What you describe is worrisome because of the risk of what you have being a sore throat caused by a bacterium that, if untreated can result in dangerous complications, such as spread to the tissues of the neck.
On the other hand, this could be infectious mononucleosis.
Sorry, I understand your insurance problem, but the bottom line is that you should definitely go back for a reevaluation that should include a white blood cell count, a full throat culture rather than a strep culture that can be falsely negative, and testing for Mono.
Good luck.