Don't give up. My wife and I send our lottery application in on Feb. 1 (I know, early doesn't mean a thing), applied for a weekday and had only a party of two. We were unsuccessful. Our window to go was extremely thin. With her vacation time, we could only go on August 31, September 1 or 2.
We were diligent in checking. Every time I got on line, I'd check. Unbelievably, August 31 opened up and we snatched it up quickly. Got lucky but that's what it takes. Don't just wait till May 1, keep checking throughout April. One of your dates could very well open up and you won't have to hassle with the May 1 rush.
Good luck!