"저는 집에 가서 잠을 자요."
Translation:I go home and sleep.
서 only means "and" in the sense of "and after that".
So for the sentence "I am a man and you're a woman" you could NOT use 서 since there is no actions that happen one after the other. So you use 고: 나는 남자고* 너는 여자야
*You might wonder why I wrote 남자고 instead of 남자이고. Both are correct. I find it cleaner to omit the 이.
So 서 can work for sentences like "Come closer and whisper", since the whispering action will take place after coming closer. 다가와서 속삭여 줘
I go home and then sleep. Based on Tips and Notes, "서" is about sequence.
If people are talking about falling asleep on the train or at home, it makes the sentence much clearer to use "and then". In English we might drop "then" because the conversation is known to be about choosing which action to do first.