If you have voicemails saved on a non-smart phone, your best bet is to record them on a computer. This is a method that requires some equipment, but not as much as you might think.
Some people found success in fixing their voicemail woes when they temporarily disabled

While this app has the exact same name as the iOS app we listed above, it’s made by a completely different developer. This version includes customizable skins and the ability to record calls or notes as MP3s. Instead of having to tap a button, you can also simply shake your phone to instantly start recording a call. You can also choose to sync your stored calls with Dropbox and select particular contacts whose calls you always want to record (and store).
After setting the number on your voicemail, you must also record a personal voicemail greeting. Follow the steps below. Start the Phone app. if you don’t see the numeric keypad, tap “Keypad” at the bottom of the screen. Tap and hold the “1” for about two seconds until the phone automatically calls voicemail. Immediately tap “3” to access voicemail settings. Listen to the options. On most voicemail systems, tap “2” to start the process to record a new greeting. The current personal greeting will play; then tap “2” again to record a new greeting. Say the greeting you want, and tap “#” when you’re done. To save the greeting, tap “1.” The personal voicemail greeting has been set. Hangup the phone. How to setup voicemail using a Visual Voicemail Applications