Another one who agrees with Vivian and Ewan McGregor's wife.
The children are key in this. Life is about way, way more than sexual satisfaction. Putting that sole aspect of life ahead of the stability and happiness of your children is unthinkable to me.
Marriage is about far more than sexual desire too. But that doesn't mean you should put up with the status quo. And I don't underestimate how tough it is to be married to someone you don't desire. There's no reason why you should sleep with him if he is so unattractive and makes no effort to be otherwise. What would happen if you let him know how off-putting his lethargy is? Can you imagine finding him attractive again if he changed - became more dynamic?
The thing is, you feel - very reasonably - a massive dissatisfaction with this one, important area of your life. And it's not one you can find an easy solution to, because you don't want to be unfaithful (by the sounds of it.) But don't underestimate how draining and difficult life would be if you moved. Guilt drains, the anger and hurt drains, the judgement drains, the unhappiness you have willingly brought on your children drains. Having no one around to do all those things you take for granted is draining. You'll be poorer, more exhausted, live in a smaller house, have less control over when you see your children and what input they have day to day.
If you truly consider all these things and know you'd still rather be free, then tell him how deeply the marriage is in trouble and take it from there. Lots of people do split up and find an amicable solution. I think a lot of them kid themselves that it was for the better for the children too. But, everyone survives.
It's not something I understand, if I'm honest. I don't get the desire to trash a stable marriage just because it's dull. But people do it. You wouldn't be the first and maybe some will come onto this thread to explain the pros to you because most of us here seem to be against it and I guess you are hoping for a more balanced range of opinions. (And sorry but the opinions of an unmarried person without kids doesn't count for much here.)