That is most likely the correct answer, though there is one other possibility: find a better lawyer. While you are probably an "at will" employee (i.e. can be fired for any or no reason), requiring a contract for no other compensation that to continue your employment reduces the level of free will in your execution of such a contact, and may in fact void it. Consider: if someone puts a gun to your head (or your dog's head) and says, "sign this", it is not a binding contact. If your employment is at risk, while it is not a matter or life or death, it nonetheless reduces your ability to freely enter into an agreement.

A better lawyer would go through this with you. But as before, the bottom line may be the same -- you may wish to just stop working for these scumbags.