1. On account of the original sin and actual sins, man was doomed to die.
Q) Is this sentence grammatical? Do we have to use 'the' before original sin?
2. All have sinned and fall short of the glory of God.
Q) What is the meaning of 'fall short of'?
I would use "Because of both original and actual sins......"
Use "fallen" instead of "fall".... That sentence means that man in not perfect as God is perfect.