Here's one to get you started. Housing Office would probably be a better source, though. This company is close to base and very familiar with military requirements. I didn't use them because they tend to keep steering you back to Browns Mills and Pemberton, but they were very friendly and knowledgable on what we could afford.
Davis Realty
S Pemberton Rd
Pemberton, NJ 08068
609-894-4333
You didn't mention a rank or where you're coming from, but you're probably in for some sticker shock! As long as you're E-6 and up, you should have no trouble finding something you can afford though, but not much in the range you gave. Always ask about the taxes -- they vary greatly, but are always high. Also keep in mind your auto and homeowners insurance will probably double, depending on where you're coming from.
Not being worried about schools helps for finding a more affordable place. (That's why Brown's Mills and Pemberton are less expensive - the perception of poor schools, but a lot of base folks live there and don't think the schools are actually too bad). The immediate area around the base is the same story. But if you don't choose one of these then a 20 min commute is a pipe dream!
Cookstown and New Egypt are as you described, but are expensive and have higher taxes. Might want to add Medford, Tabernacle, Shamong to your list to look at too. But for your price range, Mt Holly is probably your best bet to start looking in. Quite a few houses on the market right now.
Good luck!
P.S. I'll keep checking back if you have more specific questions.