We are sponsoring a kid who is 22 (wife's nephew).  He has a student
visa and cannot work except at school and for minimum wage.  We signed
an agreement stating that we will provide a roof, food and proof that
we have enough money to support him.  He is gobbling up credit cards
and using them for discretionary spending against our advice.  Once he
leaves are we liable for his debts?  I have not co-signed and as far
as I know my wife has not either and he again is 22.

Since he is an adult he is... but to ease your mind... read the terms
of service... if he runs off they will just lose their money... but
they deserve it for giving credit to a poor student.

You are not his guardian.  Just there to provide food/shelter.

You are right in your estimation.  Insist that he relinquish the cards
to you and destroy them.  Maybe the card co.s won't catch that you are
sponsering him, but maybe they will.
As you can find here or elsewhere on the web, and have stated, you are
responible for his shelter and food, i.e., that he does not become a
burden on the state.
Maybe that does not mean that you would have to cover his credit card
debts, which he apparently has incurred without your signature and
obligation to them.  Hopefully someone and answer this, and if it is
certain that you do not
Maybe he thinks you do (bad!), or maybe he has just lost control, as
many, many US citizens have. That doesn't matter, just get the cards
from him.
Sure, you need to know if you are liable for his debts, but regardless
of that, he has to learn responsibility if he wants to become a US
citizen.

This is no legal or professional advice, as the disclaimer below explains.

It sounds as if he is out of control
- and probably anticipates skipping the USA

Although he is 22 years old, my suspicion is that he is a bit younger.
- well in USA acclimatization terms

Can you think of a way of alerting the Credit Card Issuers ?
- like a letter stating that you accept no responsibility for his debts ?
- once he is on the black list, he'll find it hard getting more cards
  (unless he is rather criminal minded)

I would suggest that you discuss this with your wife

One thing I would think about, is if he is maxing out the cards, how
is he making the minimum repayments ?
- it might be a variation of Ponzi, but something makes me suspicious ...

I guess you raided his room after getting nervous about his mail
- if he is not living high on the hog, he might be loan sharking

It boils down to: he is either very thick or pretty smart