My hands get a little sweaty when I'm with my girlfriend, but that's more because they get hot from being so close to someone else, since our body temperatures are 98.6F. Sweating is one of the body's reactions to nerves, because when you get nervous you start to get hot.

I've always had a problem with my armpits just plain sweating a lot during the day, if you have that problem maybe try not using too much deodorant. I use a stick, and I used to put a ton on because I figured if I put more on it would stop me from sweating so much. I found almost the opposite to be true: if I just put a little on, maybe 1 or 2 swipes, I didn't sweat as much. But I also had to use some spray because with so little deodorant my armpits started to smell by the end of the day.

But I'm sure lots of people get really nervous and thus sweaty palms/armpits from talking to someone they're interested in. If you've had a boyfriend/girlfriend for a while though and still get nervous/sweaty around him/her, then they either probably aren't right for you (since you feel like you can't be yourself around them, which would be why you get nervous) or you just have to overcome your nerves.