Mean Girls is possible. Twice I've gotten to the theater at 4pm so I was the first on the cancel line. Ask the box office where the cancel line spot is. When I was there a couple of weeks ago, it's to the right of the most right theater entry door if you're facing the theater. They don't want it by the stage door.
So I got there at 4 and the lottery started about an hour later. I submitted my name for it, but I didn't win. So the next thing was the cancel tickets which they've had both times. The manager told me they have house seats that they hold aside in case the producer needs them for last minute guests. If they're not used, they sell them to those on the cancel line. They offered me six row center orchestra at about $200. I said I'd wait. Twenty minutes before the show, they dropped the priced to 179.
On the other occasion, because I was the first in line, they sold me a lottery ticket that someone didn't want. The seat was awful. Right orchestra extreme right. I had a speaker in front of me that was deafening and could feel the drums below my feet. So if you have the time to get on the cancel line, it helps to be the first.
My main complaint about the theater is the limited leg room. I had left orchestra D7 that I ordered in advance. I'm 5'8" and my knees were up against the seat in front of me. When I jokingly asked if they could make the seats any smaller, a woman in front of me turned around and said, "yeah, at Frozen..it's worse." The theaters are taking their cue from the airlines installing as many seats as they can.
And yes, once a show wins awards, the ticket prices go up. Supply and demand. Many said they didn't like Carousel. I did. Seen it four times just to watch the dancing which I thought was incredible. The songs are hokey MGM-type movie musical songs from the 40s, 50s. But I like musicals, be it on stage or in film.
If I were you, I'd try to reserve what tickets you can and do the cancel line/lottery for MG or any other show you'd like to see. There's also rush tickets in the morning or standing room, but don't try that for MG on Wednesday. Security said folks like up the night before. I went to MG on a Tues, Thurs, and Friday.
The other Monday show is Boys in the Band. Saw it twice. Will see it again. Good luck