"I don't want to wash clothes today."
In Chinese time words and phrases are placed at the beginning of the sentence, unlike English where the position is more flexible. You could say 我今天不想洗衣服 though.
His sentence is not too wrong either. I think Duolingo should accept it too.
I still use the wrong character a lot. I ran both versions through an online translator.
要 = I am not doing my laundry today
想 = I don't want to do my laundry today
I get the feeling that 要 is closely related to the idea of certainty of what's going to happen. 我不要洗衣服 can be translated as "I don't want to wash clothes" but must be understood as "I won't wash clothes." You don't want to, and you will not. Period. On the other hand, 想 is more of a wish. You don't want to wash these clothes, but you still have to do it. You wish you didn't have to! I hope a native can come and tell me if this is the right way to understand the distinction or if I'm overthinking it