Can I devote more room to my voicemails?
Voicemail is handled by the carrier. The size of your mailbox depends on the plan &amp; features you have with Verizon, not on the free space available on your phone. 

I doubt Verizon will keep old messages indefinitely, so if there are messages that are important to you, you should save backup audio files of them. Then clear out old messages to make space to receive new ones.