Is my understanding correct that "necessities permit the prohibited" means providing an excuse for the one who commits a prohibited act, not making the act itself permissible, and is there any fault in me for my hesitation in understanding this rule?

What you have understood is correct, and there is nothing against you. You have been advised to turn away from these delusions and doubts. So, strive to rid yourself of them by ignoring them and not paying them any attention, for a Muslim is rewarded for struggling against their own self to get rid of whispers (waswas).