I worked with Apple Chat Support on this issue. They were unable to solve it and scheduled a call with "Senior Support". After talking with them, it appears that "Pick a Song" was removed as part of the iOS update. Alarms I created with music downloaded from my music library before the update still work and I plan to keep them for as long as Apple will let them keep working. I'm disappointed that I can no longer change them or pick a song for any new alarms.
I have no plans to buy tones from the tone store, but will be investigating 3rd party alarm apps that will let me create my own alarms or use songs not included in the default set of alarm tones. I still hope that Apple will restore this ability in future iOS releases.