Hi Erik,
Reboot your computer first.
Then perform a forced restart on your phone. Hold down the Home and Wake/Sleep buttons at the same time for about 15-20 seconds until the Apple logo appears. Ignore the "Slide to power off" text if it comes up. You will not lose anything.
Then connect to iTunes and instead of dragging songs, try clicking on the ones you want on the music page and then do an Apply/Sync.