Is my Tawaf and Sa'y considered valid if I performed ablution before Tawaf, then the prayer was called during Tawaf and I prayed without renewing my ablution due to the intense crowd, knowing that I have a permanent state of impurity and am required to perform ablution for every prayer?

A woman experiencing Istihadah (non-menstrual vaginal bleeding) and someone with a continuous ritual impurity must perform ablution (Wudu') for each prayer time if anything exits. If nothing exits, they remain in their initial state of ablution. If you are certain that something exited after your ablution, then you must perform ablution again after the entry of the new prayer time for your circumambulation (Tawaf) and prayer to be valid. Otherwise, your Tawaf would not be valid according to many scholars. However, if you are not certain that anything exited, then your Tawaf is valid. As for Sa'i (the ritual circumambulation between Safa and Marwa), ritual purity is not a condition for it. Some scholars hold the view that ritual purity from minor impurity is not a condition for Tawaf. If you adopt this opinion, then your Tawaf and Hajj are valid, but you must repeat the prayers you performed without ablution.