Can a get a work permit or something, or get my case fixed without having to leave the country?

If you do not have any bad criminal history, you, most likely, could get work authorization through the deferred action program. There are organizations in New York that help to prepare the documents for this program without charge or for a very small fee (but you would still have to pay the government filing fees - $380 + $85). However, what you are saying about your permanent status does not sound right; so I would recommend talking to an immigration attorney before you decide to apply for the deferred action - you would not want to lose a chance to have a permanent solution of your immigration status. Deferred action is for people who cannot get anything else. See if your approved petition can give you a better deal.

It appears your case is concern over whether due to his poor financial situation, your father who filed your green card may be unable to file the affidavit of support showing CIS you will not end up on welfare after getting your green card. The government will not give green card to foreigners who are likely to go on welfare and so may become deportable under s.212(a)(4) INA [public charge]. So, to convince CIS you are not a welfare risk, CIS wants your father to show he has money to support you by filing a Form I-864. Now, apparently lacking such proof, CIS may deny your case. If this is the situation, not to worry, Ask your father to talk to his friends and relatives who can act as his co-sponsor so that they can show CIS you will not go on welfare. Such relatives or friends must be either US citizens or green card holders who have enough money to act as his co-sponsors. The co-sponsors along with your father are signing an agreement with CIS to be responsible if you go on welfare. Such co-sponsors must be either US citizens or green card holders and must be living in this country. You do not have to live together with the co-sponsor in the same house or even in the same state.

There may be options for you. I would suggest in paying for a detailed consultation with an experienced Immigration Attorney who can guide you correctly.