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Chewy, soft cookies need less time in the oven. When you see they are doughy in the middle, that's when you take them out and let them cool on the pan rather than a wire wrack because they will keep cooking but not overcook.
Crispy cookies need more time in the oven but less time on a baking sheet. The reason for this is the more time the cookies spend in the oven, the better chance of burning the bottoms you have, so as soon as they get out of the oven you want to let them cool for just a minute or two and then put them on a wire wrack. Crispy cookies should be in the oven for maybe 3-5 minutes longer than the original recipe may call for. But make sure you keep an eye on it. The old oven I used to have would make my cookies crisp and then burn them within a 30 second time span. Your cookies will be crisp when they are golden brown on the bottom, baked around the edges and set in the middle. When you take them out of the oven, leave them on the pan for just a minute and then put it on a wire wrack. This will help halt the baking process which means your cookies wont have an overbaked taste to them. You can also lower your oven temperature if your cookies are scorching too fast for you to catch them, but you'll have to bake them for longer.
Also, shortening vs. butter does not really matter, it just depends on how your oven is. So by all means, use butter.
(If you're worried about your cookies sticking to the pan, get parchment paper or one of those silicone mats. They help a ton for cookies.)