So 22 year old male here.
I go to a GYM a few times a week, and sometimes I see this young woman to whom I felt an attraction on sight.
The first day I saw her, I slowly came to her and ask her name. I said mine afterwards and gently shook hands with mutual smiles.
Second time I saw her, I small talked with her a bit, it was fine.
Today I saw her again and wanted to engage more.
I was doing a kettlebell exercise, and during the middle I decided to come over her and ask her if I can give her company. She gave a partial node, and I kept my talk as smooth as possible - I was vibrating with excitement/anxiety, especially my face, so it was kinda difficult to talk, but I did my best to keep smooth. I told her about how "deadly" my current exercise is to empathize with her always-tired looks at the gym, and asked her if she wants a piece of my energy bar. She said she has already eaten, but I just nodded because I was too excited to have a lasting conversation!
I then had one of those "slips" when you get excited from a crush - I landed on a pedal of one of the machines, so awkward moment. I looked at her in positive shock, smiled to myself and started doing my exercise near her.
And here is a red flag - she went away to a different station (she just started the one she was on previously).
I didn't chase her, I stayed at my spot and moved on to my other stations. The one after the kettlebell was another leg exercise station. By coincidence she came and asked if I'm exercising there. I told her yes, and asked her if she wants to go first. She nodded. We switched place, and then she switched with me, I even readjusted the seat and leg pole for her. She said the seat is not right and asked for help. I helped her, and insisted that I take it more back than forward. I did so.
I tried to ask her a bit about herself - how old she is and what she is doing right now (work, university, etc). She didn't hear or respond to my question during those moments because she insisted on adjusting her station. After she finished - wandered to the previous station.

Whenever I talk with her, she sometimes seems irritated when I talk with her. And when I talk with her there's no small-talk kind of talk (I don't know how to call it) like she has with the gym trainer who's always there when she's there (he's middle-aged, so it's just platonic). But it's not all irritations, she does answer fine, sometimes smiles. She did node when I asked to accompany her. And sometimes I even catch her glancing at me.
Today when I left the gym I told goodbye to the trainer, a person I met and hesitated whether or not to say bye to the woman I met. I went on my way, looked at her and she was back-faced. Walked again and looked at her. She happened to see I'm looking at her (through a mirror probably) and wave goodbye to me and immediately told and waved her goodbye, too.

It's been 8 hours, I wish I could write it the moments I came back home but I had to go to work since by then I remembered more details.

So anyway, I sensed red flags, but there's a part of me which is still in question.
She always seems tired and I sense that so I ask her "how's it going?", "you okay? you seem worn out!", "do you want a snack bar to boost you up a bit?". She's always insisting that it's ok, that she's not tired (I feel otherwise (but still want to believe in her (if she gives a chance to get to know each other))).
Also that women are generally introverted so there's no clear idea of what is going on inside her (learned from my own senses and experiences of course, I'd love to learn if I'm wrong).
The last thing I want to do is irritate a woman, it's kinda devastating me a bit when I see such red flag signs since all I want is to get to know and perhaps develop a path towards a loving relationship.

I'd love to hear for advice.

Thanks!