He's a prick and an idiot
This is NOT a prick and idiot. It is someone who thought that his need to have children would lessen. Clearly, he didn't think that the love of the OP was only relationship he would ever need.
The OP explained that DH thought he could give up his desire to have children. OP has said that she was honest and said that she doesn't want children. The OP has tried to figure it out again ... and decided that she is sure that she doesn't want children - "still no on the children, though" - but the DH DOES want children.
It doesn't matter how may times the OP says "let's figure out how to separate, if you want children you should try to have them and find happiness". That sounds like a forced separation that is driven by the person holding all the cards.
No, your DH should not have embarked on an affair. But given that you admit that he is unhappy in a relationship without the prospect of children, and you have been "trying with him with the best of intentions and have been sad (and guilty) about the fact he's not happy".
PLEASE LET EACH OTHER GO!
You are not compatible. It isn't about the affair at the end of the day - your DH has suppressed his feelings and life's desires to be with you.
I've been literally, questioning the meaning of life and trying to make him happy and make everything work and wondering what I could do
Your DH wants children - you don't. You won't change your mind - he can't change his. Why you married each other with such a fundamental disagreement about your future life-long plans, I can't figure.
Can you both agree to separate amicably and move on?
You can continue with your life - no children. (No guilt/re-evaluation trying to make your DH happy)
Your DH can pursue a new relationship - possibility of having children.