I found a solution that worked for me for both my iPhone 5s and my iPad 4:
1) Uncheck the new songs you want to sync, along with all other songs from that artist, in your playlist in iTunes on your computer.
2) Use the USB cable to sync (turn off WiFi first on your iPhone).
3) After the sync completes, close iTunes on your computer.
4) Unplug your USB cable from your iPhone.
5) Open iTunes on your computer.
6) In your playlist in iTunes on your computer, check all of the new songs you want to sync, along with all other songs from that artist.
7) Plug in the USB cable to your iPhone.
8) After syncing, all of your songs should have synced to your playlist on your iPhone.
There must be a glitch in the new iTunes update that Apple will need to fix with another update. At least this workaround works until then. It was frustrating that it took me two days to figure out a solution, but I hope that others see this and are spared some of the misery I went through.