I don't understand the posters agreeing that this is 'controlling'. Presumably they wouldn't call it 'controlling' to ask him to stop doing anything else that physically endangered your children and put your finances at risk?
You're not controlling him, you're simply laying out what you need for the relationship to continue. You're giving him a choice. He's free to keep drinking, away from you. Or to stop drinking, with you.
He's breaking your marriage vows, by messaging escorts.
He's risking losing his job and saddling you with sole financial responsibility.
Worst of all, he's risking your children's lives. So far he has managed to get away with it - no drunken car crash or house fire he's too drunk to notice, yet.
You're not 'being controlling', you're taking control of your life and your children's lives before someone gets seriously hurt, or worse.