If you don’t want to use Python you can use Modell Builder. Model Builder has several iterator tools.
Use Iterate Feature Selection to iterate over features in a feature class (your watersheds). Then use Clip Tool.
You have to run this modell 10 times (one for each of your 10 raster). If you want to automate this step you have to add Iterate Rasters Tool.
Here are some examples of using serveal iterate tools.