I don't know that there's a big difference. It may help to call and talk to someone. Explain your situation just as you have below. The problem, I guess, w/ contacting them before payment is that if you get someone willing to make a GW adjustment, they're not going to do it until payment has hit your account. You may have to call back once the payment is made.
I would probably make the payment and catch-up first. Then, send a GW letter and follow-up w/ a phone call explaining your situation. If you're turned down, keep trying. You may catch someone in a giving/forgiving mood.