Does sin befall one who performed ablution and prayed while in a state of major ritual impurity due to the coldness of the water and his lack of knowledge of how to perform tayammum?

If you woke up in a state of major ritual impurity (janabah) and the time for prayer had come, and the water was cold, what you did was wrong. The obligatory act is to perform ghusl (ritual bath). If you fear harm from the cold water, you can guard against that by warming yourself and drying thoroughly. If that is not possible, or if you fear illness or damage, then your obligation is tayammum (dry ablution) instead of ghusl. Wudu (minor ablution) is never sufficient in place of ghusl. Tayammum is performed by striking the earth once, then wiping the palms of the hands and the face. If you are afflicted with an impurity other than major ritual impurity, it is obligatory to remove it before prayer. If you pray with it intentionally, your prayer is invalid and you must repeat it. If you prayed with it unknowingly and then became aware, there is a difference of opinion among scholars regarding the obligation of repeating the prayer. A Muslim must learn what makes his acts of worship valid, and whoever neglects this is sinful. A person in a state of janabah is not considered ritually impure in their essence.