What's missing from your post is motivation.
You would like to stop, but there is nothing pressing.
Sometimes an addict needs to hit bottom before he is truly ready to grow.
I suppose if your wife came back from the doctor with a diagnosis of a sexually transmitted disease, things would shift in a big way.
But so far, so good.

I agree with frank.lee.  We think we're all so smart and can fool everyone; make it seem like everything is OK.  We're all great husbands, fathers and lovers.
Until we find out our wives are crying and dying inside from lack of connection with the man they committed their lives to.

For me, my motivation was that the dual life became intolerable.
And the struggle was miserable.  You know the cycle -- sneaking hits of lust here and there, fighting the urge to go further, ultimately giving in, feeling duped and guilty, or just numb.  Repeat.

It's much easier to fight this battle from outside the ring.  I just need to stay out of the ring.
Once I'm dabbling in lust, I'm in the ring, it's exhausting and I'm going to get clobbered.

I was tired of being out of control.  Of having a base desire control my life and my actions.
I wanted to be whole, to be real.
Fantasies are sweet.  Like candy.  But real food is so much better.

You did write:
"On the other hand my entire internal world is in decay. I feel disconnected, shameful, frustrated and depressed. I put on a happy face for my kids but inside I am dying slowly."

Real life isn't candy, but it can be very sweet.

None of us can give you motivation.  I'm just sharing my own.
Maybe others can share also.
Maybe something will click for you.
Before something snaps.