The phrase "I have to leave." doesn't really make sense when you think of what you're literally saying either. :P
Yes it does. "I have to go away" is used the same way. "I got to go away" isn't so correct when showing a need. The first sentence shows a need to leave while the second usually shows that they were allowed to leave.
The word "have" can be used as multiple parts of speech.
S0 R U L0L
tat n0t fair i has l0ngr s3ntnce
