Question: I don't use my cell on Shabbat, it is turned off but calls still go to my Voicemail. Is this permissible? I don't have the option of turning voicemail off only for Shabbat. It seems it is either set up or not at all.

Answer: 
No sources so please CYLOR but:

The voicemail goes to a central server at your provider. When you dial in to retrieve your messages you are not accessing a message store on your phone rather you are dialing into a VM service.

Since this server/service is maintained for both Jewish and Non-Jewish customers it would be permitted to have benefit from it even after Shabbat and there is no need to delete your messages.
There may be a problem with the instruction "Please leave a message". as you are requesting a melachah be performed on your behalf. One formulation I've heard suggested is the change the prompt to "*If* you leave a message, I'll call you back" or similar wording that does not explicitly request the caller leave a message. (No sources to hand, sorry.)