You didn't say what email accounts you are dealing with. I assume at least one Gmail, maybe an Exchange account? Also you didn't specify what phone you have. Exact procedures vary by phone make and model. The following is for a Nexus with vanilla Android. Your device may be different, but at least similar.
First, make sure your contacts are saved correctly to your Google account. Login to Gmail online and check your contacts there. Remember, any contacts you have saved using the Keep Local option will not be in Gmail, they exist only on the phone. If all other Google contacts look good, move on to the phone.
Check your sync settings again.
1. Go to: Settings > Accounts > Menu (3 dots upper right corner) > Auto-sync data should be checked.
2. In Accounts select: Google > Your account > Enable Contacts sync (and enable/disable sync as desired for other items).
With luck, your Google contacts will now sync up and you will be able to save edits and additions. If the contacts fail to sync and/or you still cannot save edits or additions, try this:
1. Go to your Google accounts as above and disable Contacts sync.
2. Go to: Settings > Apps > All > Google Contacts > Force stop > Clear cache > Clear data.
3. Reboot the phone.
4. Go back to Accounts and enable Contacts sync.
Just maybe now all will be good. Think positive. It might help to get some good karma going.
Android since v1.0. Linux since 2001