"I’m just doing my best to stand a chance with her."
No you're not. You're a long way from doing your best to stand a chance with her.
If you were doing your best, you'd be getting training in man to woman social skills. And then you'd be practising these skills on at each 4 evenings per week, with you breaking the ice with at least a dozen sets per day.
So that the next time you talk to her, or to another woman you get a crush on, you'd be prepared.
With training and practise you wouldn't say "hey sorry to disturb but you go to (x) gym right?" You'd say something like "Hey, can I borrow you for a second?" with you having a prepared follow up to whatever response she has to your opener.
Don't say "sorry" in your opener. You're not a beggar and you have nothing to apologise over.
You wouldn't say "yeah I recognised you I go there as well, how are you finding gym life"
Telling her that you go to the same gym as her as the 2nd thing out of your mouth is you getting to the rapport stage too early in the interaction. You should aim to build attraction in the first 5 minutes of the interaction before building comfort and rapport in the 2nd stage from minutes 5 to 40. A good way to build attraction is to demonstrate higher value. Make statements instead of asking questions. Flirt with her. Tease with her. Chuck in little things to disqualify the 2 of you getting together. Tell a little anecdote. Or cold read her.
EG "...Hello! You seem like the sort of person that takes pride in keeping your weight down, although sometimes you wish you were more confident about your looks..."
"cool are you new to this town or are you from here?"
Do you really care if she's new to the town or where she comes from?
A better 3rd thing to say to her is "I can only talk to you for a second. What are you doing right now?" False time constraint to put her at ease. You ask that question to establish logistics and whether an instant date would be on.
"oh nice I’m from here pretty much live here my whole life, it’s a nice town, are you liking it here so far?"
I hate this! It's building rapport again, instead of getting her interest / attention / attraction. It's the sort of thing your grandmother would say. It's not the sort of thing a high calibre man would say.
Aim to cut out all the "oh nice's" from your responses. Replace them teases and flirts.
EG "Nah, you're pulling my leg! You look like a well establish solicitor, not a student." Or if she does look mature for a student "Nah, I don't believe you. I think you're still in 6th form." Said with a small smirk on your face and the right tone of voice to show that you're kidding and mucking about.