"I am full. I want to go to sleep."
No, on the contrary I'm pretty sure that's the most common way of saying it
I actually thought it was 好飽了 for ages and recently switched to 很飽了. Maybe all three are actually right?
很 and 好 are interchangeable in many cases when they give exclamatory force (very much, so, how...)
Is it necessary to repeat the subject in the second clause? I put "我很抱， 想睡觉" and was marked wrong. I believe I've seen other sentences where the subject is omitted because it's inferred. Either that, or I've been taking crazy pills. Before I go ahead a click "Report", I'd like someone with some better Mandarin chops to weigh-in.
EDIT: Yup, literally the next question omitted the subject in the second clause: "我们吃了十碗牛肉面，真饱!" = "We ate ten bowls of beef noodles. We are so full!".