You'll be fine getting rush for Chicago after 10am. The box office attendants are very kind and the seats are great!
The other poster is right about BOM, its still quite tough to get lottery tickets as quite a lot of people turn out. If you get into the standing room line as soon as you file your lottery vote you should be able to get those seats which are a real steal at 27$.
Personally I think I'd like to see shows that are newer and not such stalwarts as Chicago/BOM. This spring is plush with new plays/musicals. Great Comet from last fall. War Paint, Sunset Boulevard, Come From Away, Amelie, so much more. I enjoy seeing new work rather than long running shows and so prioritized that on my first trip knowing almost assuredly that the two shows you are looking at would be playing the next time I came. Enjoy yourself regardless of what you see! New York is so wonderful.