I have the same exact problems with razor burn/ skin irritations in the bikini area.... Mags suggested using Tend Skin ...here's the previous post on this topic...web page
The bathing suit does scream "I am trying to hide something"...but most people will probably just think you are a modest person!! It's still cute, and if you're only going to the Y...I think what's most important is that you are comfortable.
Personally, I use a womens electric razor to shave the bikini area..it's a remington, and it does a good enough job for me (not as close as a bic, but you can only tell if you get down there and investigate...and I don't think the gals in your class are going to be doing that!!) I do alot of swimming, for physical therapy, and I'm lazy, and I try to avoid extra pain. So this is my current option of choice