Hi Scott,
You don't necessarily have to schedule your exam months in advance. In fact, I scheduled mine two weeks in advance when I took it in April of this year. I could even have scheduled it sooner, but I wanted to continue to study those last few days. At the time I scheduled, there were many dates still available, although the weekend dates had already been snapped up. At least here in the Washington DC area, there were always slots available in the near term, although obviously your mileage may vary depending on where you live and just plain-old-luck. You can get somewhat of an idea by looking at the available dates on the
Prometric website
(use "Locate a Test Center"). Bear in mind that you can reschedule without penalty as long as you aren't within 30 days of the test date.
With regard to the application, you do need to have the required 35 hours of instruction. Upon approval of your application, PMI will provide you with a PMI Eligibility ID which you'll need in order to schedule your exam. They won't give you the ID with which to schedule the exam until your application is approved (if you're audited, you won't get the ID until you clear the audit).
If you want to minimize the time delay if an audit should occur, I'd suggest that you make sure that your references are aware of the their role and are willing to verify your experience. This is particularly important if you've not been in touch with them in some time. You may find that one or more of them is unreachable and you'll need to find an alternative; better to learn this early on, rather than when faced with an audit.
Other than that, plan your time as best you can, but, like a good project manager, be prepared to pivot and deal with any surprises that crop up!
Best regards,
Tracy