^^^ States handle the PE process differently. Most require that you meet all of the requirements to sit for the exam (making the exam the last requirement), whereas some allow you to take the exam early and finish the rest of the application process after you pass. A few others require additional info (like Texas) once you pass the exam.
Here in CO, you have to meet all requirements to sit for the exam, but still do an additional application review after you pass. This is just a CYA thing on their part to ensure you have a complete and accurate application.
In any case, it's best to wait until you have the license in hand because it is illegal in most states to represent yourself as an engineer if you are not licensed to practice in that state. Just remember that waiting for your license is a much shorter wait (typically a couple of weeks), but in the grand scheme of things is a very minor setback. You've already waited several years to take the exam and a couple months to get your results, what's a couple more weeks?