All right.
I've been given a pretty straightforward solution:
1.) From your Second Life Dashboard, click the Account dropdown and go to Billing Information.
2.) Simply remove and re-add the payment method of your choice to your account.
This did the trick for me. I should have tried it sooner, but the problem seemed so out-of-nowhere, I just assumed there was some kind of server-side snafu that needed sorted out.
Anyone still having problems with this, hopefully this works for you. 😄