You’ll get charged the new annual fee at the end of the first statement period after your anniversary date – ie. sometime before mid-December. So it makes sense to cancel before then. Having said that, AmEx has been known to give a bit of latitude (ie. a refund) if you cancel just after the annual fee has been paid … provided you haven’t already used your new travel credit, of course!
In my experience AmEx is one of the quicker credit card companies as far as bonus points goes. If you refer now, I reckon the points should come through in plenty of time. In any case, if my calculations are correct you still have more than 10 weeks until the annual fee is charged.
Hope this helps!