All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
If your suspicion, which was close to certainty, was based not so much on an obsession as on concrete evidence such as seeing the filth on your Ihram clothing or so, then you have performed Tawaaf with your clothing affected with impurity. If you came to know about the filth after having finished your Tawaaf, then your Tawaaf is valid and it is not due on you to repeat it. Your ‘Umrah is valid too then, and there is no problem.
If you came to know about the filth during Tawaaf but you completed it, then it should be known to you that purity from filth is a requisite for the validity of Tawaaf according to the opinion of the majority of the scholars, although according to Abu Haneefah and a narration attributed to Ahmad, it is not a requisite.
We hope that acting upon this opinion may provide a way out for you, in which case you should slaughter a sheep and distribute its flesh among the poor dwellers of Makkah.
Similarly, if your suspicion that your child urinated during your Tawaaf was so strong that it was close to certainty but nothing of the filth affected your garment, then your Tawaaf is valid, and that there was filth on the body or clothing of the child has nothing to do with its validity.
Shaykh Ibn Baz was asked about the ruling on carrying a child during prayer or Tawaaf if the child is wearing a diaper containing filth that does not affect the external clothes. He said:
"The Messenger of Allaah, sallallaahu ‘alayhi wa sallam, led people in prayer while carrying Umamah bint Zaynab . There is nothing wrong with carrying a child during prayer or Tawaaf when there is a need for that. The child cannot be safe from any filth that may come out of him. If this filth is not apparent, and rather is kept with something like a diaper, then there is no problem, and praise be to Allaah."
Allaah Knows best.