You are under no obligation to purchase shore excursions through the cruise line. So, yes, you can just do shore excursions on your own. You don't mention which ports you want to do the activities you want to purchase at. Be advised, however, the ship will not wait for you if you are late getting back to the ship. You must be back onboard 30 minutes prior to sailing. Example, if it says you are in Cozumel until 7:00, you must be back on board at 6:30 ship time.
This brings up another issue. Make sure you check the daily planner, and listen for announcements whether or not to change your watches. It is up to the Captain whether or not to change the ship time to match the local island time. If they do not, the ship time in some cases will be one hour later than local island time, so be sure you keep your watch on ship time, otherwise, you will miss the ship, and have to catch up at your expense.
2007 - Majesty of the Seas (Bahamas)
2009 - Enchantment of the Seas (Western Caribbean)
2010 - Freedom of the Seas (Western Caribbean)
2013 - Golden Princess (Alaska, roundtrip from Seattle)