No, if you cut and bale the same day, then you wrap it. With the over mature dry " straw " tops, it will balance some of the green and should do okay for wrapping. We had that same situation a couple of years ago, when it was down to about 30-40% chance, we took the chance, cut it if it was sunny that day, and then mostly tedded it out at least once, during the day....then raked and baled or took a chance and raked the next morning with the clouds and got it baled and then wrapped when we could get the wrapper. Far from ideal, but it beats snowballs in the winter; especially when there is snow every where. You guys in NY had a pretty rough winter for snow this past year I remember and constant rain this year is a real kick in the teeth. I would also seriously consider grazing as much away from the home farm so that they can come home to grass or you might get a chance to make some " combo 1st/2nd cutting. Don't discount that it could just stop sometime in early Aug also, and dry up with no more rain and then the pastures will dry up too.
As for the water situation, we bought a 1000 gal tank from tractor supply several years ago, put it on a flat bed trailer,or a hay wagon that you ironically aren't using, and put 2 150 gal tanks with the auto water float on them and ran water from the tank via garden hose to the two water troughs that were set right against the trailer. All according to how many head, you might get 1 to 10 days out of it before just pulling the trailer home to run water into it, like overnight or whatever. With the grass/grazing being green it will have a high water content and they might go through the water a little slower than if everything was very dry and temps were in the 90-100 every day.
We haven't used the tank for 3 years, but it is there ready to go, and it was like 400 well spent. Had the trailer and the couple of rubbermaid water tubs and whatever the couple of floats and short pieces of hose were. Probably about 5-550 in it all. We had to do it when the water dried up at a pasture and it was one of the few places that had alot of grass.
There are alternatives that will cost a bit up front, but then it will give you some alternatives in the future. If you have to sell a couple of head, it will pay for it and give you some flexibility.