To show extreme embarrassment, you can say, "I was mortified!"
People usually say, "I don't feel like..." when they don't have a particular reason for not doing something. Another time we might use "I don't feel like it...." is if we don't want to explain ourselves to someone else. It is a way to deflect a person's question, so even if we have a reason, we don't want to share that reason. The meaning a person intends will come through their tone of voice or facial expressions.
"I don't want to go outside" doesn't really give us an idea of why a person doesn't want to. It is simply a statement.