In fact, the plane was filling up with water much, much quicker because the vents were not closed.
I was researching this some more and actually, both things happened--the ditching checklist was not finished AND one of the rear doors was partially opened, allowing water to enter the cabin.
There is conflicting testimony by the cabin crew and passengers about who actually opened the rear door. Multiple passengers had said that it was one of the flight attendants who opened the door. That same flight attendant says a panicked passenger pushed past her and started to open the door. An NTSB hearing was convened to answer a number of questions, including this one, but it seems to have been inconclusive regarding this issue.
In any case, icy cold water entering the cabin from that door made it more difficult for the passengers in the rear of the cabin to make their way forward to the overwing exits.