I’m not on Verizon, but my carrier does use their network.

Did you buy your phone from that carrier? Because if you did, then it’s a Verizon model (because it’s carrier locked and running on the Verizon network), and will have a permanently locked bootloader. You can check this by going into the phone’s Developer Settings and finding the “OEM Unlocking” toggle. If it’s greyed out…Sorry bud.

According to my carrier’s support pages they’ll automatically unlock the phone a couple days after it’s paid off

Hopefully. But please keep in mind that SIM unlocking (i.e. locked to a carrier) is not related to bootloader unlocking. Far too many people get confused over this concept.