I am positive your Goodwill has at least "some" all cotton shirts, unless it is very tiny--but their shirts are are kind of pricey.
Just keep looking. Consider it a "hunt". I wonder where you live. Every thrift store I've been in has at least a few all cotton shirts. And I've been in quite a few thrift stores in my day.
But if you aren't careful you can spend a lot of money and you might as well just order from fabric.com or take a look at hancocks-paducah.com. You can spread the word among your friends and/or your husband's friends (or other couples if you have no husband) that you are looking for cotton shirts. You may get given a bag or two of shirts - don't be surprised if about 75% of them aren't all cotton - just pick out the ones you can use and pass on the rest. It takes a bit of time to collect shirts like this.
Or try Craig's list as well and freecycle.
Here's an auction
http://www.ebay.com/itm/Lot-of-8-Men...4AAOSwc1FXX3aX