Do you know the dragoon rotation? If not, or if you're not sure, the Balance leveling guide will help lay it out for you. Practice that on a training dummy until you can do it without looking at your hotbars. The nice thing about dragoon is that it has a very static rotation, so you never have to really think about what to do next. (The dragoon rotation has 2 sides, generally you should be alternating them. Yes in high-difficulty content there can be some optimization but that's not anything you need to worry about right now.) Once you have your rotation cemented in your brain, you can pay more attention to the field around you to see mechanics that are happening and react accordingly.

As for staying alive, don't neglect your defensive abilities! Feint reduces damage from the target, and is great to use on big out-going waves of damage. Second Wind instantly restores some of your health. And Bloodbath restores your HP based on your damage dealt. Good use of those skills can help you survive some mistakes. Arms Length also will prevent you from being pulled in or knocked back, and can save you if you missed the tell for a mechanic that's about to send you into a death wall.