Not trying to discredit the fact that today's situation was a bummer - totally valid.
If I could offer a small ray of hope though:
While the TodayTix lottery is, unfortunately, the only way to get cheap tickets to this show. But if you're ok paying full price, then I wouldn't recommend relying exclusively on it. In my experience, no matter how popular a show is, and no matter how sold out it is, there will almost always be scattered tickets that open up down the line. I've gotten into quite a few sold out shows in my day, through 1 of 2 main method:
(1) Persistently checking a show's ticket website to look for returned tickets. Especially the week of. This tactic has been very rewarding to me, on many occasions. (and I see above that this has already started happening for Merrily)
(2) Cancellation line / waitlist. Again, in my experience, even the most popular shows have a few cancellations per performance. If you're 1st or 2nd in line for that day, there's an extremely good chance you'll get in. Easier said than done, I know, but if you can plan ahead for it and carve out the time on a day off, it's possible. And almost certainly better odds than the lottery.