Thank you for your interest.
Yes Cruise ships do have stabilisers and they do their best to avoid the nastiest bits of weather. (4,000 people being seasick at the same time just isn't much fun.) And there is only one cure for seasickness. (Sit under a tree. )
The VLCC's I sail on can cope with most weather, but there is a big difference between when they are loaded and when they are in ballast. It takes a lot to make things uncomfortable on a loaded tanker, but when unloaded we are so high out of the water that saying is: "They roll on wet grass". We don't have stabilisers or anti-roll gear so things can get rough. The only good thing is that being so broad in the beam we take very long, slow rolls. (And yes, we are so long that the fore end can be rolling to port while the aft end is going to starboard.)
Basically even the biggest most stable ship cannot beat Mother Nature. If you really hit bad weather, head into it and do the slowest speed that will keep steerage way. I got stuck behind Hurricane Katrina in the Yucatan Channel on a 250,000 ton ship. Just sat there for 3 days, doing half speed and going nowhere, until she decided to move. Poor old New Orleans.
Plenty more tall stories where that came from!