I think he drinks because he is such a troubled guy. In the first part of the story, we mostly see Henry as Clare saw him when she was young. There he seems more stable and wise, probably since he is an adult. However, as the story develops we find that Henry is more of a troubled guy who isn't comfortable with time traveling.
Additionally, the roles switch so that Henry doesn't know the future but Clare does.
Didn't Henry tell Clare when she was younger that he wasn't supposed to drink, because of doctor's orders?
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