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Unclear if you're starting from Southampton - if so - nice weather from Maderia/Azores onward. If you're flying to the caribbean, then november is a lovely time to go. Hurricaine season should be long past and the climate is always gorgeous as are the sea temperatures. Lucky things!
We did the Caribbean last November and the weather was great. It depends on whether you're doing eastern or western. The eastern caribbean season really starts about October time so you could still get some showers, however these are usually few and far between. The benefit you've got is it's still warm with it.
The sea conditions should be fine. I have cruised for about 20 years and prefer the caribbean as can I still get a bit seasick and it's normally really smooth seas. You will obviously get a little bit of movement when you're leaving a port but we've never experienced any more than that.