I live in a city where there is no sound of azan and only countable masjids that recite azan only within their area of masjid. I wanted to ask if i have to wait for the azan to happen in the masjid even though I can’t physically hear it to be able to read Fajr? Or can i read it as soon as I finish praying tahajjud and it is time for Fajr? I have been doing the latter just wanted to confirm what is the ruling if you can’t physically hear the azan do you still have to wait even though the time of Fajr has started? And is it any different for men or women?

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The timing for any Namaz has no relation with azan that azan has said or not
Each prayers timing is different based on when the time for any Salat has set in then it’s mustahab (recommended) first we say azan and then iqamah and then we start our Salat.
Therefore if you can’t hear azan it doesn’t meant you can’t start your Fajr Salat. The best thing which we all do in the western world where it is not permitted to call Azan on the loudspeaker, we cam have an azan app in our mobile phone so if you have fixed it according to your city it will start automatically calling azan right before every Namaz.
Also there is no difference in Namaz timing for men and woman.