Is it safe to send official GREs almost on deadline (e.g., the day before)?
I am in the same predicament. I asked Princeton Admissions about this and they said:

We continue to match scores to applications as we receive them, even after the deadline.  If we already have your scores on file when you submit your application, it can take up to seven business days for the scores to match.  If you recently took an exam or recently made a request to the test administrator to send the scores to Princeton, it can take up to three weeks from the test date for us to receive your official score report and electronically match your scores to your submitted application. You will be able to monitor your submitted application checklist and will see the test date once the scores have matched.  You will not see the test scores.