I think that's a question of style. I would use "go" for the action but "be" for the state.
If I make one more payment from my current account, I'm going to go overdrawn.
I couldn't buy the shoes I wanted because I was already overdrawn.
I agree that the situation differs between current (checking) accounts and credit card accounts. My current account has an authorised overdraft of £300. However, if I am already £300 overdrawn and I use my debit card (checking account card) to make a payment, the payment is likely to go through. I will then be more than £300 overdrawn, which will class as an unauthorised overdraft and I will incur heavy charges.
My credit card, on the other hand, has a fixed limit of £4000. As far as I know, if I already owe £4000 and I try to make another payment on that card, it will be declined. I won't be able to use the card until I have made a payment which takes the amount owed to below £4000. I would still only be able to use the card again if it meant that the amount owing didn't exceed £4000.