How can I handle dry-looking areas in the table I'm refinishing?
Good question.  My thought is a little too late for you, but I have found that some woods excrete "oils" for a long time.  You can usually see these after sanding, but it requires a close look, not only from the top but also across the surface.  If you see anything different from the rest of the surface, then you can try more sanding.  If that does not work, then you need to seal the whole surface until the problem area is no longer visible.  When finishing, my first and second coat are usually 1/2 product and 1/2 thinner.  I find this helps soak into the wood and acts like a sealer.  If there are spots then you need to address them before proceeding.