The valve core could just need tightening. You can pick up the tool at any auto parts store for less than $2. I have removed the core and it is no different than any other. You can either pick some up at the parts store as well, or a tire store would probably just give you one.

What you describe sounds like it may just be the core. I would try this first before going for the rebuild kit and having to break the tire down to install the kit.