I had a surprisingly difficult time digging up info on this topic via search so I'm wondering if it's a silly question, but...how do I get a new IP for my Linode? I just setup CloudFlare and I'm going through their protecting the origin steps, which involves changing your IP once you've patched up all the ways someone might discover your true IP. I see that I can purchase additional IPs for my Linode, but is there a free way to just get a new one?

Pretty sure you can't get a free ip. You might be able to delete your existing Linode and redeploy to get a new ip and sort out CloudFlare first.

You can always open a ticket with Linode. The worst they can say is no. Make sure to explain that you want to replace your existing IP with a different one, not get an additional IP, and be sure to explain to them why you want to do so (for CloudFlare).