What is the ruling on doubting that something has exited the private parts during wudu’ or prayer due to obsessive thoughts, and is it obligatory to purify the area and repeat the wudu’ and prayer every time, or should these thoughts and liquids, which may be natural (vaginal moisture), be ignored to avoid increasing the obsessive thoughts?

The cure for whisperings (waswas) is to ignore them and not pay them any attention. Giving in to them corrupts one's religion and worldly life.

You must perform ablution (wudu) and pray without paying attention to whisperings about the emission of prostatic fluid (madhy). If you are afflicted with excessive vaginal discharge and do not know when it stops, then your ruling is that of a person with incontinence (sahib al-salas): you perform ablution for each prayer after its time has begun, and vaginal discharge is pure.

If you doubt whether it is prostatic fluid or vaginal discharge, nothing is required of you, for the religion of Allah is ease.