Their refusal to give you a copy doesn't make it unenforceable - but their reluctance suggests that they have doubts about its enforceability.Quoting davepull
The absence of witnesses, or the absence of a signature or initials on each page, is unlikely of itself make the contract unenforceable. Those are practices which can make a contract easier to validate or enforce. (e.g., the initials on every page show that no pages have been switched, and a witness can attest that the employee's signature is authentic.)
It is my understanding that under Florida law, a judge won't enforce a restrictive covenant unless the employer demonstrates a legitimate business interest worthy of protection. An employer cannot impose a non-compete agreement merely to punish or unduly burden an employee who wants to quit. Florida also permits an equitable defense based upon the employer's bad conduct, and it may be that you would be able to defend based upon their post-agreement cutting of your benefits, late pay, and refusal to give you a copy of the agreement.
If they seek to enforce the agreement in court, you would be able to obtain a copy through the process of litigation.

