'I have seen the word ‘obligated’ used in an occupational context'

That definitely does seem to be a place where obliged does not seem quite strong enough.

"To feel "obligated" has a connotation of being morally required to do something you don't want to do. To feel "obliged" has a connotation of being morally required to do something you DO want to do. "

That's just simply not true. As far as I'm aware, the word 'obliged' can always take the place of the word 'obligated'. However, the reverse is not true.

'Obliged' can and does certainly mean to feel constrained or coerced into an action.

Or somewhat subtle in difference that you feel responsible

It can mean you feel indebted to someone. However, this can be appreciative 'I am very much obliged'. Similarly, when using this positive spin on the word, you can oblige someone.

There are a number of other ways you can use the word too.