This is a business, not your buddy you owe money to. A company can not help you out just because you're going through a rough time, they operate on a business level, not a personal level. You pay money to have a service provided to you. Your contract you signed CLEARLY STATES that you must cancel by the 5th of the month in order to not be billed for next month. That's the reality of a professional business. Do you expect the same leniency from your utility company? Do you call them and complain about how loyal you've been, and expect a delay or for them to stop your billing cycle early just because you're going through rough financial times?
In regards to your claim that the executive office just wants to charge you an extra month to get more money:
Of course it doesn't take 30 days to cancel a membership. Once again this is a business relationship, things must operate on a business cycle. That date is the 5th. By your own comments you clearly understand the policy, but just expect some special provisions be made for you. Or, what about someone who is not going through rough times but just simply wants to cancel their membership, should you be given special privileges not given to them just because of your hardship? Once again, do you feel that you should recieve a special discount on your morgage just because of rough times.
As far as you cancelling your credit card: Let me save you the trouble, instead of cancelling just declare bankruptcy, because that's the only way to opt out of a payment you are legally obligated to make. What Life Time will do if you refuse to let your card be charged is simply have one of their lawyers pull up the contract YOU SIGNED saying you must cancel by the 5th or be charged an additional month and send it to your credit card company. If you still refuse your legal obligation they will send you to collections.
The contract you signed clearly stated the cancellation policy. Sorry you're going through a rough time, but that doesn't magically alter your financial obligation.