I'm not much of a sports gambler myself, but it's always interesting to hear from those who are. Congrats on your winnings! Even if they weren't enough to trigger tax forms, it's still important to report them as other income, as you mentioned
My father had the same problem. And he had a hard time. We were forced to remember and write down all of my father's trips out of the U.S., plus we had to bring a document confirming all of those trips out of the country and the trips back in.