Do an encrypted backup with iTunes, and then just don’t wipe the old phone until after you restore the backup to the new phone and have verified they’re still there. No big deal.

Also, I’m fairly certain they are actually stored on your carriers servers anyway, and not the actual device. Thought I’m not entirely certain. That’s definitely how it is for non-iPhones.

Also, I don’t know what carrier you have, but here is n example from T-Mobile of how you can download your voicemail messages from them: https://support.t-mobile.com/docs/DOC-1171#fifthheading

It should be fairly easy to find similar instructions for other carriers.