Since you're either unable or unwilling to repost, I'll just have to speculate on what you meant.
Since most of my points have pertained to 4th and 5th Amendment issues and you specifically mentioned the 5th in one of your replies, I assume you are saying the tests violate the 4th and/or 5th Amendments.
Allow me to explain why you're wrong.
One of the conditions of receiving these benefits is that the recipient cannot be currently using illegal drugs. If you believe that being drug-free should not be a prerequisite for receiving benefits then that certainly is a viewpoint that is worth arguing. However, the method by which the state of Tennessee is weeding out (wink, wink) drug users is not a violation of anyone's constitutional rights, so long as the process is being administered as described.
If the state required all applicants to submit to a urine test then that would be a constitutional violation. If the state required random applicants to submit to a urine test that would be a violation of constitutional rights.
Asking applicants to swear they are not using drugs is not a constitutional violation. When I renew the license plates on my car, the state requires me to sign a paper swearing that I have liability insurance coverage on the vehicle. When I submit my income tax return, the government requires me to sign the return and swear that the information is accurate to the best of my knowledge.
I don't have a constitutional right to lie about having auto insurance. I don't have a constitutional right to cheat on my taxes. And welfare recipients don't have a constitutional right to lie about meeting the drug-free requirement for receiving benefits.
If being drug-free is a requirement for receiving welfare benefits and a potential applicant is a drug user then he has three options. He can simply not apply for the benefits for which he is not eligible. Or he can be honest and acknowledge he has used drugs and hope he passes the urine test anyway. Or he can commit welfare fraud and lie about being drug-free and hope he doesn't get caught.
Asking a welfare applicant to swear they are drug-free is no different from asking me to swear I have auto insurance and that my tax return is honest and accurate. Asking a welfare applicant to swear they are eligible to receive benefits is not a 5th Amendment violation.
And so long as the only applicants who are urine tested are the ones for whom there is probable cause to believe they are using drugs then no one's 4th Amendment rights are being violated either.