If I reset network settings will I lose pictures and contacts?

Resetting network settings on iOS should not effect those items, however if you have mixed up the terms you may be talking about something different.

Can you try explaining where you see the 'reset' option

e.g. is it one of these is so which one?

Settings app > Wifi > …

Settings app > Mobile > Mobile data network …

Settings app > General > Reset …

It would be a good idea to make an iTunes or iCloud backup before making any changes.

If Apple suggest that for the issue you should try it.

Voicemail comes over the telephone data network, perhaps if you reset & recreate the settings it could work correctly. I suspect you will need to contact your carrier if it continues, sadly this is something that could be because of either party so Apple may pass you to the carrier who then pass you back to Apple…

Maybe book an appointment at an Apple store if you have one locally.

Good luck.