Hello!

It's great you've joined the flock!
We have put a couple of alfalfa/brome square bales in with the flock during winter. In my case I haven't spread them because I use the bales to add another dimension of height. I've had a case of the chicken math as well...although not quite as plentiful as your situation. I may/may not have the area required for the number of birds in the coops for winter...lol. So I've adapted to using vertical height in the coop to help.
The birds like to pick at the more tender parts of the hay; the grass leaves, the alfalfa flowers and tender shoots. The base stem of the plant tends to get left and gets added to the bedding once they eat it down to the point the bale is loose and loses shape.
The best alfalfa hay would be the cuts you could get before the hay gets very tall. I see you're in New Jersey. I'm guessing people get more than one cut of hay. Here we get one cut..maybe 2 if we're lucky and our hay the last few years has been coarse because it rains when we want to be cutting it and our cut has been delayed.
Yes, if you are able to keep it covered, out of the sun and out of the rain...the hay you buy will still be good. We are just finishing up our hay, both round and square. The round is placed in rows and the squares are put into the loft of the barn.
If you've had rainy weather, you'll want to make certain the bales you're buying do not have mold in the centers; that's both round bales and square bales. It usually shows up as whitish or black colored hay. Either the baled have been baled too green...or in the case of square bales...they've been allowed to get wet through the bale.