Because you are the one who leased the apartment, you could be liable if your husband does not move out. Perhaps you could give him your own notice now. Or, if he does not pay you any rent, just tell him to leave.
Yes, since your husband is there with your permission, he is a lawfully there. If he doesn't move out, the landlord will have to evict him. Doesn't matter if it is you or your ex, you are on the lease and ultimately responsible for the guests you allow to stay there.
He will be evicted. and yes you could be liable for any money judgment -- unless you get your husband to indemnify you now. another idea is to get the landlord to agree not to seek judgment against him only. or you could move to evict him yourself while you still have a right of possession ...
Responses to Avvo questions are based on a general discussion of the law and in no way constitute legal advice. No attorney-client relationship is created, and you should not take or fail to take actions based on my answers. Consult an attorney in your area. Time is often of the essence. Act quickly!