First off, are
your requirements the same as
your wife's? If not, please mark this as a dead thread.

Some people that I know of (and some I know) have done away with the traditional 'dining room' and 'formal living room' and used those areas for bars, pool tables etc. This could be an option if you have rooms like these that don't get any use. And, if your family is 'cool' with it.
It's a shame to waste that < 6ft basement area though. Regardless of the stone, I imagine it could be waterproofed for a price. Excavate the exterior and waterproof from there (not sure about the floor). If your family is 'small', that space could be used for kids' rooms, I'd guess, freeing up other areas?
If you go the 'addition' route, must it be with a basement? If you can build an addition off the main, could that be your game room?
Also, don't be too quick to rule out a bigger house in your town due to pricing. Housing prices are down, even in NJ. If you've got the time, it can't hurt to contact a Realtor and give them your specs/requirements and let them do the work for you!