Fellow UK-er here! I've done lots of cruises out of the US and always booked everything separately, never had any problems (touch wood) and been able to pick dates/flights that suit me.
My only suggestion would be to fly in at least 2 days early if possible - that first afternoon on arrival can be a bit of a write off with travel, tiredness and fighting jetlag, and secondly, if your bags don't make your flight for whatever reason, they can be put on the next day's without you having to worry about arrival time. Always take some spare clothes, meds etc in your hand luggage just in case. Slightly less of a worry with direct flights than if you have a connection, so it would probably depend on where you're flying from.