Large project "shifts down" subsequent spreads when I edit ones before. How do I edit individually?

If the text is all in one thread, there's not much you can do to move text "downstream" without affecting each of your following pages. If that object moves down one line, the next page will too, and the next, and the next ... until you reach a spot where that "one extra line" is assimilated into your existing layout.

There are a number of ways to account for your layout changes, depending on the type of document you're working on. Is it a book layout you're working on? Then it can be easily accommodated by adding a few extra pages to the end of the book. Is it a magazine feature in the middle of your issue? then you may have to crop a photo or cheat your text slightly to accommodate it.

If you can let us know what kind of document you're working on, and maybe provide a screen capture of how your layouts change when you use your return/enter key to move objects down on the page, it'd give us insight into how best to help you.

Randy