You probably already know about rush policies, but in case you don't, rush, lottery, and standing room (SRO) are almost exclusively the ways I use to get tickets to shows, since I, like you, cannot afford full-priced or even discounted tickets to most shows, especially Broadway and Off-Broadway.
Basically, all three of these are bought on the day of the show. For more "unpopular" shows, tickets will be easier to get than they are for more popular ones. In the case of Miss Saigon, I know that they offer rush tickets for $39 each. So, what you're going to do is go to the box office in the morning (ideally close to the time it opens, or maybe even a bit before, in order to maximize your chances) and then ask for two rush tickets for that day's performance. If they have any available, they'll sell them to you, and then you're set!
I don't believe Once on This Island has announced a rush, etc. policy yet, but keep an eye out!
Enjoy your trip! I'm sure you both will have tons of fun!