Men weren't mortal because they worshipped Morgoth, they were born mortal.
Elves were the First Ones, and the ones that survived the original trip to Valinor weren't perfect by any means. I'm Jewish so "original sin" doesn't mean much to me theologically, nor does it explain to me how some elves were goody two shoes and some were evil assholes.
Well, I can try to explain it using the story of Adam and Eve (which is, after all, original from the tanach)
Adam and Eve are seduced by the snake (whether you believe the Christian retcon that it's Satan, or not, is pretty irrelevant for that) and are punished with all that stuff about painful childbirth, and toiling for the fruits of their labour and well, being doomed to die.
In Catholicism this act is referred to as Original Sin. The first time someone sinned, and it's believed to be the root of all evil. It believes that even babies are already born with this sin because of what Adam and Eve did.
In Middle Earth, the original humans didn't bite in an apple, but instead they were seduced by Morgoth into worshipping him instead of Eru;
https://tolkiengateway.net/wiki/Tale_of_Adanel
So Eru punished them with
shorter lifespans, feebleness in old age, susceptibility to sickness and greater susceptibility to evil and greed and violence and all that stuff, and even the world itself turned against them, which is why they can't do things the elves do, such as walking on freshly fallen snow without sinking. They also lost their ability to desert their bodies (i.e. die) at will, which is also something the elves can do, because the fear of death (which in Tolkien is the root of all the evil men do) forces them to cling to life no matter how degraded it is.
And because every single one of the original humans was seduced into worshipping Morgoth, all of their descendants, all of humanity is fallen under this doom.
The original humans were, apparently, meant to be more like the Numenorans, living very long lives and then deciding to voluntarily join Eru outside of the physical universe before they decline.
The idea seems to be that the Numenorans for their role in fighting against Morgoth in the War of Wrath, had some of their doom removed, but fell back under it when fear of death and greed for immortality got hold over them again.
Originally this mortality was the great gift of humanity, because it meant their souls could leave the world and be with their creator. Elf souls on the other hand are trapped in Ea and either have to go to Mandos in case of death or become feral, bleed into nature and become susceptible to being used in necromancy.
That's because the original Elves never ate the apple, never were seduced into worshipping Morgoth. So in the Biblical version the Elves are still "living in the Garden of Eden" without toil or sickness and being less susceptible to the temptations of evil as a people, and being able to leave their bodies in cases of great peril (hence why the whole thing with the Orcs didn't work anymore, Elves would voluntarily die and flee to Mandos before being debased like that or being forced to take part in procreation)
So Elves have a natural inclination towards what Tolkien saw as moral behaviour because they are still in an unfallen state, but can individually choose to do evil, while humans are much more tempted, and have to choose to do good.
The problems this concept created with the already established storylines of the Silmarillion is one of the reasons why Tolkien never managed to create a final, full version of it during his lifetime.
(sorry for the novella, I hope it makes sense)
