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 iPhone with Apple (for use as an iPod touch only. with no phone service) 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 had given me earlier, 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.

The Sprint SIM card model that **did** work with my iPhone 5C was Sprint part number SIMGLW406C.

The Sprint SIM card model that **did not** work with my iPhone 5C was Sprint part number SIMGLW406R.