Sigh, I had a nice reply typed out and my computer ate it...
However, I agree that the close gym sounds like the better option. Here's the thing, it's pretty easy to avoid just one person at a gym if it really bothers you. After going there regularly for about a week or so you can probably figure out this person's schedule and roughly know when they'll be there and just come at a different time! If they do show up just go to a different part of the gym.
BTW, I was also nervous when I first started going to my gym and I was BY FAR the biggest woman there. Not anymore, though, because I kept going despite my insecurities. At first I would pick the bike towards the back of the room and stick to this little stretching area that was walled off from most of the other people. Eventually I got over my fear and just dug right in. NOBODY CARED.
That's the biggest thing, I was the one with the insecurities and if anybody did care they aren't the kind of person that I'd want to associate with in the first place.
Now I'm no longer the biggest woman there and I love seeing ladies bigger than me work out because that means they are doing something for there health! Gyms are a place to make people healthy and the reason you might not see too many overweight people there is because if they keep going and make changes in their eating patterns too they probably won't stay overweight for long!
If you still feel a bit unsure of yourself, why not join some classes to start with so you can get a feel for the place and you might even meet a lot of people who are supportive of your weight loss journey. They might have even BTDT themselves!
Oh, and I agree with others. Anybody you know from school is going to know your size already. It doesn't matter if your huffing and puffing too because they'll see you working hard trying to change your body. That's a great thing!
ETA: oh, and even at my very skinniest when I was a HS/college athlete I always sweated A LOT!!!! My weight/athletic status had absolutely nothing to do with how much I would sweat!