Great work giving yourself an opportunity like this. Joining a gym can have lots of benefits, from improved health, to being part of a community.
In relation to feeling uncomfortable attending the pool and going for a swim, what are you looking for in terms of utilising your membership? Are you familiar with swimming? If so, lap swimming might be a great option, and it might help you to ask someone at the pool or a personal trainer how best to organise a routine, to help you with the ambiguity you might be feeling.
If you are less comfortable, or looking more for something more social, you could enquire about swimming clubs, or perhaps other sports, such as water polo, that the facility might offer. This could be a great way to help move outside of your comfort zone.
Finally, I commend you for reaching out for help at all, and the Beyond Blue forum is a great place to start, but have you discussed what you are experiencing with a GP or Psychologist? It can seem like a big effort to organise such things, but it can be well worth it.
And of course, a pool is a public place, you are more than entitled to use it, even if you might feel "creepy". Having a routine will help giving you a sense of purpose though, which might reduce this feeling.
Good luck, I hope you manage to get started soon.