a little late to join this conversation, but here's my thoughts...
something to think about... since you're probably not looking to bring a lot of stuff into the country, finding a furnished place can be harder... the 1500US mentioned above can *probably* get a somewhat decently furnished 3bdrm appt in a respectable area (won't be modern furnishings)... i'd personally up it to about 1700 just in case, but i have not looked in mobay, only kingston and mandeville... can find stuff for cheaper, but the areas you're looking, probably wouldn't want to.... at that price you'll be getting a good community, gated/controlled access, good sized place, properly wired and good running water (maybe even luck out and get hot water

)
don't rule out kingston completely, there are safe areas that have some great new complexes going up... will likely be way out of that price range, but you just may luck out finding a rent from owner type unit in a nice area.... living in the city does have some real advantages over living outside it (i'd personally rather live in say hope pastures than anywhere in spanish town or even portmore... but i digress)...
in a city area at that rate, you'll probably be fully wired for cable/phone/inet (all probably necessary with american kids and a work from home situation)... phone/inet will run you about 3000J a month, cable is cheap at around 1200J a month... both reasonable...
current varies, but will likely run you around 3000-5000 J a month... if you need an airconditioning unit you'll double that or more... i can't remember what water bills are, they may end up being included in your rent... maintaining a cell in jamaica as you probably know is pretty cheap, less than 3000J or so will get you a 1200 min to US and some local time... could be more or less depending on how much you talk locally...
other things to think about, groceries and maintaining a vehicle in jamaica can both get pretty costly (esp with american children who might want the comforts of home when it comes to dining)... i can't help a lot on how that will impact your budget but provide my end figure below for comparission, i'm a relatively spoiled person with reasonably spoiled children... i didn't want to sacrifice too much at their expense so i did make my estimates with some wiggle room for comfort
all and all, for my family of (soon to be 5 of us) i budgeted around 3500USD a month living expenses in total and had it blessed by a few people down there who said its def doable... that's not including one time payouts (car, school, medical appts, airfares etc)... with one less adult, you can probably knock a few hundred off of that, but who knows how much that "on again/off again" one will be in your life if you're down there
a little cost savings can be found in having your children share a room (you might have ruled this out because you have boy/girl?)... might not be as socially acceptable in north america, but it wouldn't be as much of a taboo on the island... 2bdrms are easier to come by and a bit more reasonable.
can you live in jamaica for cheaper?, yes, absolutely... people down there do it on 500US a month or less... but for someone looking to give their children the safety and comforts of home, well it's worth the investment to make it a good living by jamaican standards... keep in mind that once people get word of you coming into an area and being "from farien", well, sadly you may become a target... (that's why a stand alone home wouldn't be recommended, townhouses or apartments in gated/controlled access areas would be to your benefit and for that, you pay)
my estimates may, by others standards, be completely off, but i spent a lot of time thinking about this over the last few months... we've since decided to move in with family and use the rent money to help fix/build/develop the house, it's win win for all i think... so i'd like my research to help someone somewhere

hope it does
