10 m resolution has nothing to do with the Z values, just the X,Y cell size. what are your Z values in? if they are in meters, then contour the data using the metric equivalent of 20 feet
You could also use the "z-factor" parameter in the Contour tool to do the unit conversion (for example, from a DEM in meters to contours in feet). From meters to feet would be a z-factor of 3.2808 and of course the inverse would be 0.3048.