Does broadcloth get softer after washings? Thanks

I think that broadcloth is what most button up shirts are made of and would soften the same way.

broadcloth is either cotton or a poly/cotton blend. the cotton broadclothes soften up with washing...the poly blend really stays the same...for ever!
broadcloth is a heavier fabric than quilting cottons and usually used for pants...home decor projects...crafts where a heavier fabric is needed. it pairs well with denim for heavier quilts.
i've never seen a broadclothe shirt...for clothing broadcloth is usually used for pants, skirts...items where you want a heavy- durable fabric.
the best selection of colors of solids seem to be in the broadclothes.

It feels like an oxford type men's dress shirt does but comes in cotton or poly/cotton blend so if you want all cotton--you have to check the fabric content label on the end of the bolt. I would put it in the category of shirtings but it's a slicker feeling cotton than shirtings. I think it must come in different qualities (thread counts) also because I have seen some broadcloth you can read a newspaper through and others that are more like bed sheets. I have used it with success and failure. Before I knew the difference, I had a poly/cotton blend run like pantyhose when a new sharp needle cut a thread. One of my first quilts and I had specifically chosen a certain burgundy fabric without paying any attention to whether or not it was a blend. Now, I always look!

There is a section for "broadcloth" in my local hobby lobby - it is nothing like what ckcowl is describing - all poly-cotton blends, pretty thin. More like what others have said, what button down shirts are often made of.

Before I knew better I used some in a quilt. It was a light color, and you can really see the seams where they overlap. Also my machine doesn't like working with it much, for some reason.

I've heard that there is such a thing as all cotton broadcloth, and I'd like to get ahold of some for shirts for my husband, but they sure don't carry it anywhere local to me.

This sounds more like Twill or Poplin than broadcloth. Gabardine would be the permanent press poly/blend version. At Joann's, they are in the "bottom weights" section. They are a nice heavier fabric to mix in with denim. The heaviest 100% cotton broadcloth I have ever seen was like Chambray but not quite as heavy--almost, though and it came in different colors. I made maternity blouses with it. This was 20 years before I became a quilter but it would have made excellent quilting fabric.