How long after baking do American chewy cookies get their normal texture?
The cookies will set (take on their final texture) about by 20 - 30 minutes out of the oven.  Coming directly out of the oven, cookies will absolutely be soft and squishable; which is why you should wait a few minutes before moving them off the baking sheet and onto a cooling rack (you can cheat this time if your cookies are all on a sheet of parchment paper, and you move the parchment paper as a whole).