If you don't mind a little commute (which really isn't bad with a fac/staff parking permit), look for a place in the Highland Park/North Brunswick/Edison/Milltown areas.
I spent two years in an apartment in Edison (on the border of Highland Park) called Valley Manor. It was a one bedroom, ~600 sq. ft., with heat/hot water/cooking gas included for around $900 a month. It was a pretty good deal when I took into account the area and online reviews and such. The only other bills we paid were for a phone line (RU offers free dial-up access for off-campus students), cable TV and electric. My roommate and I rarely used the AC and were pretty good about shutting things off - our electric bill usually ran around $30 a month.
This year I'm in a place called Maebrook at Renaissance, in North Brunswick. I share the area with 3 roommates, all grad students. For ~1600 sq. ft. we pay $1,615 a month plus all of our utilities. Phone/Internet/Cable is around $100 a month and Electric/Gas usually runs around $130.
New Jersey is defintely an expensive place to live, but you can get by here. My fiance actually has the same aid package you were offered - tuition remission and an 18k stipend - and he has no financial problems. He has to keep an eye on his budget, but he's never short on cash when he needs it.
If you have more questions about the areas or anything, feel free to email me through my member profile.