Thought I would drop by. Your sitch is difficult however not lost. Your H is trying to maintain a lifeline to you. He is doing this with his E mails / Txt messages etc. During this time he has his OW ( and do not mistake this it is an OW ). He finds this new life exciting and seductive but is not sure that its what he wants long term. He will be very confused by this , probably for some time. He will be guilty and at times will try and blame you for what he is doing to ease his guilt.
Where you need to get to ( and this is a journey nothing happens overnight ) You need to detach from your H . This does not mean not caring about him but to find activities and a life for yourself that does not depend on him. ( I think you know this ). It does mean starting to cut the strings he is hanging on with , and that is not responding to his txt messages / emails unless they are about the kids or some other logistical need. Being happy when he is around , not because he is there but just because.
Even if you dont feel like it start acting like you are getting on with your life , take no interest in what he is doing who hes with.
Make any time together pleasant and enjoyable but in a way that you would be if he was not there.
Your actions will speak louder than your words. Once he feels like you are letting him go , he may just start to consider the consequences.