A darn good lawyer. It sounds like you were accused of possibly taking patient drugs... A urine test could have cleared that up, unless you have your own at home and sometimes take them as many do... We are only human and have many aches and pains - which means you should show them your prescription and that would prove you don't need to risk taking them from anyone - you have your own! I don't know your whole story, but you would have to prove you have not been able to get another job - which would be part of your damages. You'd also have to prove stress and other - so you'd have to show you've seen a doctor for it..
The facilities are usually very large and have great lawyers. Your best bet would be to get your name cleared and then some compensation from them. They sometimes might setttle to get rid of you. You should probably take that. Some get really ugly and will try to destroy you. It can be very expensive and you don't want to go through it with those a_ _ h_ _ _ _ _. You really don't.
Again, your lawyer has to be great. I doubt any would take it on a contingency basis. There might be ALOT you can do with a legal assistance firm. They could help you file your own case. I'm not kidding. You wouldn't have to pay a lawyer a dime. You're the one with all the facts. You might find a friend who could help you with this. If things got further along - like depositions, trial, then you will need one.
Get the paperwork for filing an EEOC complaint. You write your story - wrongful termination, loss of wages, loss of enjoyment of life, embarassment, shame, depression, fear, stress, palpiations, insomnia, damaged reputation, ect... Then the employer will respond. See what happens after that. You will be referred to a mediator and you could settle for something. (Part of that settlement would be to clear your name - I hope.)