To clarify: neither Dolly nor Hamilton have a "rush" policy. Hamilton has a lottery, as well as SRO (?) and the cancellation line. I haven't followed what the situation is, but I gather you'd have to literally camp out for at least one night to have a shot, or have things gotten easier since a few months ago?
Dolly has SRO and as far as I know people start to line at 5-6am.
Anastasia I think also has a digital lottery, not a rush, so you'd need a lot of luck since I assume lots of people enter every day for a handful of tickets.
Great Comet has both a rush and a digital lottery. For the rush I think you'd have to be in line 7-8am for tickets, depending on the day.
Dear Evan Hansen has a digital lottery and SRO and from what I've heard it's similar to Dolly, in that you have to line up very early.
War Paint is the only one of those shows with an actual rush policy, and I think being in line between 8 and 9am should secure you a ticket.
I suggest you search for the respective topics on this board and check out recent reports, especially closer to your dates. The Tony nominations might will likely impact some of those shows, i.e. they get crazier with lots of nominations. If you want to be on the save side or not wait in line for 5+ hours in the middle of the night, some of these shows have discounts on Playbill, Theatermania ect., namely War Paint and Anastasia. Great Comet hasn't extended its discounts beyond May 21 yet (except for the dates were Dave Malloy is in for Josh Groban), but I'm hopeful they will soon-ish!