If the problem is with your phone, the factory reset has a good chance of solving it. If not, contact your mobile service provider, to carry out some troubleshooting to help resolve the problem from their end.
Most of the time, your iPhone voicemail is full because the voicemails you deleted on your iPhone are still being stored somewhere else. At the end of each voicemail, press the number designated for deleting voicemails. This will erase the messages saved by your carrier and free up space in your voicemail inbox.

That’s not to say strip all emotion and personality out of your voicemail, but be yourself! Presumably, you are leaving a voicemail with someone because you want to continue a conversation with them either by phone or in person. What will they think if the “personality” of the voicemail doesn’t match the one in more fluid forms of correspondence? Knowing how to leave a voicemail that is equal parts succinct, professional and courteous can be hard for those not accustomed to doing so.
If I’m not available, just leave me a voice mail. Again, either way, it will be good to know what’s going on. Thanks in advance for that, and I’ll look for your call. You can reach me at (your number). That number again is area code (your number). Thanks ________.”