Apparently Apple is able to update the phone's profile on their end and change the carrier. Turns out I didn't need to restore after all; inserting the SIM card into the phone somehow restarted the activation and the phone switched carriers from Verizon to Sprint.
But before I did that, Apple recommended I restore the phone so it would switch to the right network. I didn't want to do that because iTunes would only let me restore to 6.1.3, which I didn't want to do because I wouldn't be able to jailbreak afterwards.
At one point, the senior advisor that was handling the issue suggested I jailbreak the phone to switch the carrier
. The phone already was jailbroken and in the midst of looking for unlock software, I popped the SIM back in (inserting it earlier did nothing for some reason) and it switched the carrier.
Long story short, I was able to activate the phone without a restore, which was good news for me. Thanks for the replies!