Try Scotch Plains. There is Bayberry (Rivervale Ct) (cats are OK). I know Duncan Hill is expensive. I heard negative things about Country Club Village (Scotch Plains). Hillcrest in Clark (Sweet Briar Drive) was fine a few years back, not sure how it is today.
I believe some of the newer apartments will have washer and dryers, but they are expensive. There is a new apartment complex in Garwood (North Ave/cross from Shop Rite) but it is expensive. On Rt 22, in Watchung (near a Shopping Center Target/Stop and Shop..etc) - I am not sure how much the rent is at that location.
Nicole, not sure if this is a possibility for you, but you might want to consider renting a small condo. That way you're only dealing with an owner about the pet issues instead of a management company that might have hard and fast rules. Just a thought! Good luck with your search.
I'm somewhat familiar with gsmls, but if you want to expand your search to Middlesex, www.middlesexcountymls.com is a great site, and lets you search for "Residential Rental." Just did a countywide search now and there were "more than 100" found for rents under $1500, and 42 for under $1000.