Is one sinful for delaying Fajr prayer due to repeated vomiting and diarrhea, and fear of the illness and cold worsening?

If you experience a wet dream and find discharge, and you are also suffering from vomiting and diarrhea, you must perform ghusl (definition) (ritual bath) before Fajr prayer. If you fear that cold water will worsen your illness, then heat the water. If you are unable to do so and fear harm, then perform tayammum (definition) (dry ablution) and pray on time. However, if you did not have a wet dream, you only need to wash away the impurity and perform wudu (ablution), without ghusl. In all cases, do not delay the prayer beyond its prescribed time. It is not permissible to delay the prayer beyond its time. If you can purify yourself with water, then do so; otherwise, perform tayammum. If you can wash part of your body but fear harm from washing the rest, then wash what will not cause harm and perform tayammum for the remainder.