I own real property free and clear. I would like to leave it to a cousin who is married but make sure she never has any claim to it. Can I do this?

I'm going to assume that the "she" you refer to in your question is your cousin's spouse.  If you leave the property to your cousin outright, his/her spouse will probably have some sort of claim to it after your cousin dies, and possibly before.  However, you can leave your cousin a life estate (meaning that your cousin would have the right to use the property during his/her life, but won't own it) with ownership passing to domeone you designate (your cousin's child perhaps) after your cousin dies.  This will prevent your cousin's spouse from having any claim of ownership to the property.  However, if you don't leave your cousin full ownership, your cousin won't be able to sell the property or mortgage it.