After numerous calls to Verizon Wireless, resetting the phone, and having Verizon calling me thinking that I had changed the ringtone, I figured it out. It's not convenient, but it does work. You have to change EACH contacts' ringtone. Open your contacts list, then select EDIT, scroll down to the ringtone and you will see that it has the DEFAULT ringtone set to it, even though you changed it under settings. Select the ringtone and then click on change and you can change it and save it. Then when you receive a call from that contact, it will be the ringtone you chose instead of the Default.
I could not find any way to change all contacts at once. They need to make it so that when you change the sound settings it should keep what you selected, but it's just not working.
But changing EACH contact definitely worked for me. I was very close to returning the phone because it would not do what I wanted it to.