You can find good 2-bedroom apartments in Brooklyn and Queens for $2,500-$3,500, which would get you within your budget if you had 3 roommates. As far as finding roommates, your best bet is definitely Craigslist.
If I were you, I’d look for something in central Brooklyn (Crown Heights, Bed-Stuy, Clinton Hill, Fort Greene, etc). You’ll be within about 1:15 of LGA and JFK and within two hours of EWR, all via public transportation.