Praise be to Allah.
Your ghusl is valid, and you do not have to do anything, in sha Allah. The fact that you rinsed your mouth and nose after you had done ghusl and dried your body does not matter, because it did not go against doing the actions one after another in ghusl, and the time between doing ghusl and rinsing your mouth and nose, is the time it took to dry yourself. This is regarded as a short time that does not cancel out doing the actions one after another. Moreover, in the case of obligatory ghusl, it does not matter if rinsing the mouth and nose are delayed until after finishing ghusl, because doing the actions in order in ghusl is not obligatory.
For more information, please see the answer to question no. 88066 .
However, we may note here that there is a considerable difference of opinion among the scholars as to whether rinsing the mouth and nose are stipulated as part of ghusl. As you asked someone who is specialised in fiqh, thinking that he was qualified for that, and he advised you based on a scholarly view that carries weight, there is no blame on you for acting on what he said. In fact, this is what is required of you, even if it becomes clear to you later on that his view is incorrect. What was done was valid, and you do not have to repeat it.
Note: it is not a condition of fasting being valid that the person should be in a state of purity whilst fasting, as has been explained previously in the answer to question no. 181351 .
Based on that, if someone did not do the obligatory ghusl, or he did ghusl but it was not valid, then he fasted, his fast in that case is valid.
And Allah knows best.