You don't "make it work".
Legally YOU are not married since it is against the law to be married to more than one person at a time.
If your husband "intentionally" deceived you then you shoud walk away. If on the other hand he "really thought his divorce was final" then he needs to do whatever it takes to make sure the divorce becomes legal. After that you can re-marry him in Las Vegas or in front of the justice of the peace.
As it stands right now if he got hurt you could legally be kept out of his hospital room because you aren't family.
If he were to win the lottery tomorrow his "real wife" would be entitled to half that money. If she ruins her credit it could have a bearing on his credit which could affect your lives together.
The list goes on and on.
Most states I know of require a certified copy of the divorce decree before issuing a marriage license. I'm not sure how this "slipped through the cracks"