Unable to add new contacts

About three months ago, my old phone spontaneously removed all of my contacts. I was hoping upon getting my new phone that I would have my contacts restored, but this has not been the case. I have tried transferring my contacts from my iCloud account, but it has not worked. Only some of my contacts are in iCloud, which is fine. I don't' mind going through and re-adding the rest of my contacts. The issue is my phone will not allow me to do so. I tap on my friend's number and hit "Add New Contact." It will let me enter all of their contact information, but as soon as I hit done, it doesn't save any of it. It just returns to having an anonymous profile and their number. I can't edit my card in contacts either. I'm not sure what's going on, and I would appreciate some help.

Thank you.