I solved the problem by making an appointment at the Genius Bar at the local Apple Store, explaining what I was trying to do, and asking for their help. Even though I purchased a used iPhone 5c from a third-party, they brought out a working new Sprint SIM card and activated the phone for me. They also let me keep the SIM card in case I need it to reset the iPhone.

The new SIM card that the Sprint Store gave me, for one reason or another, did not work. 

The Genius in the Apple Store told me that Sprint SIM cards are very specific, and some would work in a 5c and others would not.