I also started out with Tesla as paying by Cash. I then asked my SA to change to financing after I had a VIN. Once you have the VIN, you will be able to download a MVPA. Do not start the DCU loan application until you have a MVPA.
I actually signed up for a DCU checking/savings account a couple months ago anticipating I was going to apply for the car loan with them. This way, it made the auto application quicker. I believe you need to be a DCU member to apply for a loan and qualify for the additional 0.5% rate discount. You can sign up to be a DCU member and auto loan app at the same time, it just may take longer to complete (a few days vs. 0-1 days).
Since you are on the east coast, don't stress out about timing. Delivery will be about 2 weeks after a VIN is assigned. Giving you plenty of time to obtain financing, rate matching, and trade-in. If you live in California, then delivery can be pretty quick after VIN assignment.
I was on vacation when I received a VIN on June 2nd and 6 days later I have completed all of the needed steps with over a week to spare before delivery later next week.