Is it permissible to combine the intention of making up for missed Ramadan fasts with the intention of fasting the White Days during the days of making up, and then complete the make-up fasts afterward? And is the fast valid if one drinks water at the beginning of the Fajr adhan, believing that Fajr has not yet clearly begun?

It is permissible to fast the Ayyam al-Bid (the White Days) before completing the make-up fasts for Ramadan, according to the preponderant opinion. As for your drinking after the Fajr adhan began, thinking it was not the Fajr adhan, there is a difference of opinion among scholars regarding the validity of the fast on that day. The view that it is valid has strength, even if the majority opinion is contrary. If the fast was supererogatory, then it is hoped that you will receive the reward. But if it was obligatory, then it is prudent to repeat it to avoid the scholarly dispute.