Becca:
Wet a paper towel with mineral spirits and hold it in one hand.
Fold your sandpaper and hold it in the other.
Sandpaper under the work, MS-dampened towel on top, With the lathe on, work the two hands back and forth in unison.
The towel will pick up any discoloration and wood dust before it has a chance to be ground in by the sandpaper on the next rotation around the blank.
This also has the advantage of keeping your sandpaper from loading so your sandpaper lasts longer.
Wipe the blank along the grain (lathe off) between grits.