Can I use my Rosary if the crucifix fell off?

Yes, you can still use it, does not matter if it is “incomplete” what matters is how you use it. You have to remember to say the prayers proper to the part of the rosary that aren’t there anymore. The Apostle’s Creed and other prayers before getting to the circle part of the rosary.

Another thing to remember is to meditate properly whilst saying the Rosary.