Im learning about getting up in the morning and learned that one must wash their hands before anything else otherwise they will violate the prohibition of baal t'shaktzu?
What is baal t'shaktzu?
I think you may be mixing up going to the bathroom and washing your hands. Orach Chaim 3:17 discussing someone who is witholding themselves from going to the bathroom says they are commiting the prohibition of Bal Teshaktzu. I am not aware of anywhere that says that one who does not do Netilas Yadayim is committing such a prohibition.
Bal Teshaktzu means it is disgusting or unpleasant behaviour.