Hi Rick,
Hopefully the following will give you the result you need:
First up you need to create annotation for your contours. - Esri does have a contour label setting in maplex that you should use to get the best results.
Next take the annotation you have - and create masks for them. You may need to experiment on a distance for your masks, as you don't want to block out to many contours in some locations.
Remember to mask only your 'placed' annotation - other wise you might have masks blocking contours where you have no annotation.
Add the masks to your map and use advance drawing settings to mask the contour lines.
An alternative is to use the masks to clip your contours - but the advance drawing options seems to work fine.
Hope this helps,
William.
ESRI link below should help:
Labeling using the Contour Placement style—Help | ArcGIS for Desktop