Recovering lost voicemails after carrier switch

Luckily, this was possible (my important voicemails were not lost!) and you have three options, which I'll describe below. The first two will require a Mac (or maybe PC would work) and will cost you money (if you haven't already purchased the right app, or find an alternative that's reputable and free), while the third may be free.

The best option: obtain from iTunes backup

If you recently backed up your iPhone to iTunes on a Mac (fairly recently, before switching carriers) then you can use an app like PhoneView ($30) or iExplorer ($40) to recover your voicemails from the old iTunes backup. PhoneView is free for up to the last 10 voicemails, but for more than that you'll need to buy it. With iExplorer, you'll need to buy it for any number of voicemail exports.

PhoneView:

• Connect your iPhone to your Mac (be sure to trust the Mac) and open PhoneView.
• Click the "iTunes Backups" button in the top bar of the app and then click "View Voicemail iTunes Backups".
• Choose the correct device in the Phone dropdown, then select all items (with command + A) and choose "Save Messages" to save them to your Mac.

iExplorer:

• Connect your iPhone to your Mac (be sure to trust the Mac) and open iExplorer.
• In the left sidebar, click the dropdown arrow next to the most recent backup of your iPhone.
• Choose the "Voicemail" button you'll now see in the sidebar.
• Click "Export All..." to export all of your voicemails to your Mac.

The other good option: restore from iCloud backup

• Check in iCloud settings to ensure your phone was last backed up to iCloud before switching carriers.
• Then, factory reset your device (or, better, a separate iPhone whose data you don't care about losing).
• Once it resets, set it up by signing into your iCloud account and restoring from an iCloud backup. Be sure to use the most recent backup (before switching carriers) of your iPhone.
• Wait for it to restore, then connect it to iTunes on your Mac. Follow the iTunes portion of these instructions to create an iTunes backup of your newly-restored iPhone on your Mac. You don't need to encrypt the backup.
• With the iTunes backup made, start following the instructions in the section above ("The best option").

The questionable, but maybe free option: call the GLDC

I contacted AT&T's chat support to ask them what to do before figuring out the methods above. Their representative told me to do the following:

To recover your VMs, you'll want to call the Global Legal Demand Center (GLDC), formerly known as the National Compliance Center (NCC). Their number is 800-635-6840.

Just ask them you need your VMs for legal purposes.

Kind of questionable, but maybe worth a shot!

Hope this helps, and leave a comment below if I can help more.