Is my fast valid after I drank, thinking that the dawn (Fajr) adhan had not yet been called, only to realize later that the Fajr adhan had been called 4 minutes before I drank? And was it obligatory for me to verify the adhan?

As long as you drank after the dawn broke, your fast has been invalidated. If it was a supererogatory fast, you are not obligated to abstain [for the rest of the day] nor to make it up. However, if it was an obligatory fast, you are required to make it up, according to the majority of scholars. You should have verified the time and not relied solely on the information provided by a young girl.