Contract Law: Is a Contract Binding if It's Not Countersigned

My question involves business law in the state of: New York

We are a recruitment business and we've signed the terms of a Company who is looking to hire and started working on their open vacancies. However, they have not sent back a counter signature for the terms. What are the risks for us as a recruitment company?

There are clause on there that restrict us, for example, from headhunting their employees but if they do not countersign and we do pull someone out, they can sign it on the spot and claim they have a fully executed document?

Contract Law: Is a Contract Binding if It's Not Countersigned