Yes you can remove the ink stain out of your cotton shirt very easily. First of all you should not wash your shirt if there are dark stains on your shirt. You need to remove them first so that you can easily get rid of that stains so take some liquid remove and make the solution of the stain remover which you can use in order to get rid of stain from your shirt.
So just make sure that you have remove all the stains form your shirt and then wash your shirt. First of all you should take a towel or a cloth and dip in into the solution you have made in order to remove the stain from your shirt. So with the towel or piece of cloth rub the mark of the stain on your shirt slowly so that it can transfer on your shirt easily.
So once you are able to remove the color of the stain from your shirt then you will be able to get rid of that stain easily. So this is the way to remove the stain from your shirt.
I use the Spray & Wash brand, but I hear Oxiclean brand does just as good job removing stains. All you have to do is spray the stained area, rub it in, and give it a few minutes before you put it in the wash. Or use the same steps with what ever (hopefully some anti-stain) detergent you have, give it a few minutes before you put it in the wash. I hope it works for you.
What I would suggest you do is to buy a washing bleach for clothes. Mix the appropriate amount of the bleach with water in a tub. Follow the instructions on the back of the bottle. Soak the clothes in the tub for the stated time on the back of a bottle and hopefully the stains will go away.