Some one at my job spread rumors that I came to work intoxicated, smelling like alcohol. This promted a supervisor to come out at my job site to investigate. After talking quite rudely to me, and telling me she was going to take me down to the office for a sobriety test in front of clients I have worked for for 2 years, coming to the conclusion that I HAD NOT been drinking, she left the job site with very stricked warnings for me. Not only did my clients question if I was a drunk, but, the entire office assumes I am one too now. Can I sue for slander and defamtion of character?

Are you ready to find a new job... because if you don't have a contract... they can fire you for any reason... even they don't want you around any more. That's how right to work, works.

If you have a union go talk with your shop steward.

There is no contract, I am a full time nursing assistant. I travel nurse.

Opinion here.. I'm betting you really need the job. If they fire you , you won't have money to pursue a suit. It won't be quick or cheap, or certain.

Every so often you have a boss that is having a really bad day and goes off, The best boss I ever had in 31 years of working... has had days like that on rare occaisons, one I have great respect for... and maintain contact with even though I haven't worked for him in years.
Its usually in your best interest to suck it up and deal with it... unless it becomes a frequent occurrence. If its frequent... then it's a different thing.

You have to decide what's in your best interest rather than are you right or not... being right means little if it results in being unemployed in an economy such as this.

... can i sue for slander and defamtion of charecter?

No.

As I understand it, she didn't accuse you, she investigated a rumor. And if you can find the rumor-monger, I doubt that you could successfully sue him/her either. Allegations of intoxication would not be defamation per se, so you would have to prove specific damages. I don't see that.