True and False. Short term, closing a CC, in and of itself, will never decrease your score. However, if you close your very last open CC you could impact your mix of credit and that can drop your score. But if you have other open CCs aside from the one you are closing then there's no impact with your mix.
If you close a $0 balance CC and if you carry balances on your other CCs, then your util will increase and that can hurt your scores too. If your balances on your other CCs are under 9%, and you close a $0 balance CC, then there's really no point damage.
If you close a CC with a balance, then the balance and the CL will continue to factor into utilization for as long as there is a balance reporting. Once the CC reaches $0, then both the $0 balance and the CL are removed from scoring.
Closing a CC will never hurt your history. Your AAoA and length of history is preserved for as long as it reports. However, closed CCs continue to report for 10 years from the date of closure. Ten years out, it will be removed from your CR and that can impact your score.
IMO, PIF ASAP. Economics comes before your FICO score too.
DO NOT GET SHAFTED BY CITICARDS---OPT OUT! I did the exact same thing about 2 months ago and I was worried SICK that it was going to crash my score. I owe 14k to citi at 7.24%---I could in no way afford or ever want to pay 29.99% interest so I had to opt out. It hasn't affected my score at all because it is still carrying a balance--just like the orinigal post questions. Do not worry about any sort of impact on your score. IMO, opting out and closing the card could make you look like a better credit risk because that balance can only go one way in the future, down! So, please opt out before you run out of time and good luck getting the new house!
A solid answer if ever I've heard one. It doesn't get much better than this...