In some cases you might save money, but there are some advantages to booking the excursion through the cruise line. Especially if you are traveling across the island and a distance from the dock, if there is an accident or something that prevents you from getting back in time, the ship will leave without you unless you have booked an excursion through them.
We did excursions through the cruise line (Princess in our case) on all 3 of those islands and were pleased with our choices.
We did not book any cruise ship excursions on our last cruise. We tried to book reliable independent tours to get aways from the crowds. Some we paid about the same as we would have with the cruise line, but it was worth going with smaller groups.
My favorite excursion of my cruise was Silver Moon in Barbados. We were on a sailing cat. with 3 other couples. The moment we stepped on the boat they started serving us drinks and kept them coming. First class service! We stopped at 2 snorkle stops - a ship wreck and the turtles. We had better snorkling, but the Silver Moon crew made our trip.
Antigua - I can't really help you there. I can only not recommend Eli's Eco tour.
St. Lucia - We went to an all inclusive resort "Rendezvous". I think we paid $40.00/pp that included all drinks, food, and beach toys. It is a small resort, but is was nice and still cheaper than Sandals.
On Barbados we did the smiw with the turtle tour booked through Princess. My wife never wanted to snorkel but wanted to see the turtles. She's not a convert and loves snorkeling. I've heard good and bad about how to book. We always use the cruise line excursions for places we haven't been before. For convienience and safety. If we know the area, we book on our own. If you book on your own, alow PLENTY of time to get back to the ship.
How many travelling with you? On many of the islands you can arrange tours with the drivers yourself at more reasonable rate than the ship's tours.
You just need to be mindful of when the ship leaves, as they will not wait for you.