Three rejection letter for internship application, one is for citizenship and other two for capability. Not easy to find an intern or job in US for international students, it’s real. That reminds me about the story of Xu Wei who had been depressed for many years, Wentworth Miller was rejected over 500 times for audition, Jeremy Lin who was only backup in Golden State Warriors.
Rejection is quite common just like sales persons try to persuade people for 100 times and succeed for one or two times. Rejection means more try rather than beating you down, what you should is to make up and become competitive.
Time is tight, adjust emotion, go ahead!