'generally obliged' is correct and usable in written English.
It is used to describe a feeling of obligation. For example, "I am generally obliged to help my neighbors."
In his later years, when Farmer--like other prime players of his generation--was generally obliged to travel as a soloist, working with local rhythm sections, he continued to make efforts to expand his musical horizons.
Then, the Europeans generally obliged.
And they have generally obliged.
American banks are not generally obliged to collect beneficial ownership information when establishing a business relationship.
As he has yet to find anyone who shares it, he is generally obliged to be patronising.
Even with a confession, however, prosecutors are generally obliged to provide detailed evidence that shows how a suspect committed the crime.
Candidates and presidents generally oblige them, Barack Obama and Mitt Romney included.
Most cameras in the city are private — installed by businesses or homeowners — and when the police request footage the owners generally oblige.
But while a cable system must generally oblige all of the local broadcasters that demand carriage, a local station is not obliged to give its signal to the cable operator.
When pollsters ask people for their opinion about an issue, people generally feel obliged to have one.
Thomas accepted these often flimsy narratives as a form of apology; in his limited experience, beautiful women did not generally feel obliged to make excuses.
