Your best option is to talk to your landlord. If you have a portion of the money that you could pay you might be able to make an agreement with your landlord to pay and stay the additional two weeks, but it your landlord isn't obligated to allow you to stay unless you pay the entire amount due on the judgment. If your landlord won't agree to allow you to stay, protect your relationship with them by explaining your financial circumstances, moving in an orderly way if you can, don't wait to be evicted, return your keys, and leave the apartment in good condition.
Being evicted is costly and emotional, so whatever you do, move your belonging out before you are evicted even if you have to put them in storage, sell some items on craigslist or similar sites, or leave items with friends or relatives. Be especially careful that you remove important documents, like Social Security cards, birth certificates, bills, transcripts, etc., and irreplaceable items like photos and children's items. Contact social services and see if you can get emergency rental assistance and look into emergency housing options. Facing eviction must be extremely difficult, but minimize the long term consequences by protecting your belongings and looking for emergency housing until you can find something permanent.
The Maryland Legal Aid Society (http://www.mdlab.org/) may be able to offer you more advice and help, as may local Baltimore social service agencies.