Too soon? No.
I moved from Placer recently to El Dorado. I scheduled my initial appt once I knew my closing date. I scheduled it for the week after closing. I immediately changed my DMV registration and got a new one issued. When I had my first appt to turn in my application I only had my registration and my mortgage booklet. They could not accept the booklet, but started my process anyway. I then supplied the 2nd form of residency at the interview. DMV wouldn't issue a new drivers license with my address.