Thanks Von.  What photo editing program do you use where you got the arrows so nice?

I attached the hoses as tight as I possibly could, with pliers.  The washers are brand-new (came in the package with the new hoses). 

From your arrows, it appears as if the water is coming out of the area with the red arrow.  When the hose isn't attached, the water runs out of the spigot just as it would as if the handles weren't completely shut off. 

Glad you think this is a simple fix.  Do you think that fixing this setup make my washing machine work too?