Here's a solution that's likely not optimal, but worth a shot. It requires on average 3N/4 submissions, or N submissions in the worst case. To begin, submit random answers and get the number correct - we'll call this configuration of answers the baseline. Now, go down the list of questions in pairs, and changing your answer for exactly two questions and resubmitting.
On average, 25% of the time, both answers were right originally and your score will go down by 2, and 25% of the time, both were wrong and the score will go up by 2. In both cases, we now know the answers to both questions. The other 50% of the time, you'll have one question right and one wrong with no change in score, so you need to go back to the baseline configuration and change only one answer.
On average, it will take 2 submissions plus the baseline to get the right answer for half the pairs of questions, but only one submission plus the baseline to verify that the other half of pairs is correct already. This gives us an average of 3/4N submissions, since you don't need to worry about the last question (if you don't have a perfect score by then, change the answer). In the worst case, however, we need to change every single answer, resulting in N submissions.