Does this sentence read well?
Everybody knows that heavy drinking and smoking combined result in a greater damage of vital organs.
I would drop the 'a' and 'combined'.
Everybody knows that heavy drinking and smoking result in greater damage of vital organs.
'Greater' is linked to 'heavy'. They, too, could be dropped to give a much simpler sentence:
Everybody knows that drinking and smoking result in damage of vital organs.
Fist of all, it should be damage to vital organs, rather than of.
Omitting the "a" is correct. Omitting "combined" is probably a good idea, but it may change the meaning slightly. The original sentence may be claiming that the combination of heavy drinking and smoking is more damaging than either drinking alone or smoking alone. Or it might just be careless writing. After all, the writer says greater damage but does not specify greater than what.
Your last sentence is the best and clearest of the lot. But it also makes a stronger claim, namely that any amount of drinking damages one's organs, not just heavy drinking.