The clay is by Crayola and is somewhat soft(?) when dried, what product would be best to use for attaching it to my face? I do not want it to be ruined as I worked really hard on it and it took me about 2 days to complete.
That material is not likely to survive very long. Normally you would make the piece out of latex or silicone, because those materials are resilient. However, I can recommend that you use rubber cement. It is generally (unless you are allergic, which you can test by putting a small dab on your skin, if it itches or get red, wash it off immediately with warm soapy water). Assuming you are not allergic, now try to attach another sample as a test to your face. Figure out how much glue and how strong the clay is. Do this by applying the cement to the back of the clay and immediately attach it to your face.
If it works, then do it with the final piece. Good luck
Spirit gum is the classic adhesive for makeup work. This time of year you can get it wherever Halloween costumes are sold. The rest of the time, amazon probably carries it.
As for improving your creation's durability, I would suggest painting it with several layers of skin-friendly non-oil-based makeup paint. Allow this paint to thoroughly dry before attempting to glue it on.