Both finally and eventually mean 'after a long period of waiting, something happens'. However, eventually also carries a second meaning of projecting into the future, i.e "I will eventually get married and have kids." That makes its usage a little more ambiguous. If you want to emphasize that something has taken a very long time to happen, but has happened, use finally. You can sometimes use eventually, but not always, so just stick with finally until you can naturally 'feel' when to separate them.
Some examples:
He finally found a job -- After a long period of searching, emphasizes duration and uncertainty of results
He eventually found a job -- To me it says that his finding a job was never uncertain, all it needed was time
I eventually want to be a doctor -- projecting into the future, what one will become one day
I finally want to be a doctor -- We don't say this~
I hope that helps, even though it's still a bit unclear. In your essay I corrected, it needed to be 'finally' because you were emphasizing duration, and also the uncertainty of 'when exactly' the government would give out money.
Cheers!