Is there a way to cycle between full screen windows on OS X?
The short answer is no. OS X, by default, does not have a method which allows you to contextually change between spaces, staying in the current application.

From my experience, your best bet is likely to focus on tabs for your navigation, replacing (or augmenting) spaces. Safari tabs can be switched between with the keyboard using Command+Shift+{ and Command+Shift+}, which would give you functionally the ability to quickly switch between pages, while still not worrying about other applications getting in the way.