As other posters have said, aside from the fact your husband is certainly lying to you and possibly cheating, he sounds like an aawful husband,
You've taken the whole weight of your family on, and it appears not to have impacted him at all. You've stopped working, with all that entails now and in the future, particularly financially for you; you've taken on all of the children's care - children who need additional care and home scholling; you've given up all of your social life; Your life revolves around keeping your family happy and safe and looked after.
His life appears not to have changed. He's kept his career, his friends, his social life.
He must be uniquely stupid to think this is fair, or uniquely selfish if he knows it isn't fair but doesn't care.
Even if there was no issue of another woman, this is not a good basis for a marriage. One person working all the time and getting zero support or downtime, while the other has mates, social events, golf, and a career to build on.
And then there's the OW. He's lying to you, you know that much. Some posters have suggested that he may be lying because he knows it would give him grief if he told you the truth, but that doesn't make the lying OK. It means he knows he's doing something that upsets you (and which if he's got half a brain he knows would upset anyone) and yet he'd rather continue doing it than stop.
Lying to you. Meeting the same woman, secretly, this time in a secluded place.... yeah it doesn't look good.
The fact he has time to go for romantic walks (or even if we give him the benefit of the doubt, a friendlt walk) with another woman while his wife never gets downtime is shocking.
When was the last time he suggested a lovely walk in the countryside with you? Or suggested that you meet a friend for some downtime?
And to then suggest you've been unfaithful. Not only is that offensive in the circumstances, it suggests he has been, or is planning on being unfaithful. He's projecting, it's on his mind, he's trying to convince himself that you've done it to him so he can do it to you.