You'll likely find more selection in Silver Spring than Rockville or Bethesda, but unless there is a reason for targeting just those areas, I wouldn't hesitate to look around. There are some very nice areas in northern DC and many in Virginia as well that are convenient commutes to downtown if you plan it right. One sort of out-of-the-way site that you might poke around in is
military-by-owner.com. You don't have to be military to use it, but you do need to pick a nearby military base. Bethesda Naval Hospital should be good for the MD suburbs, the Pentagon or Ft Myer for VA, and probably Walter Reed for the north side of the District. Not sure what the supply will look like at 4+ BR's, but almost all of it will fit under a $4K cap, most of it comfortably so.
Renting for a while is a good idea if you don't mind moving a second time. It's hard to get it just right until you have a feel for things. Not such a good idea is that of driving into downtown DC when Metro is an option. Figure the cost of gas etc. plus $20 per day for parking (though you could cut into that some with a monthly contract), then figure that many employers (including all government employers) participate in Metro's SmartBenefits program which pays (tax free) for up to the first $110 of your monthly mass transit (but not parking) costs. Bottom line? Metro is much faster and cheaper, and a lot less stressful than driving. It just makes no sense not to use the train when it's an option.