I completely disagree with those that have said that they will see the "truth" some day. That's not necessarily true. They will see what they see!
I don't know all of your story but what I do know is that if you don't speak up, further on down the road (and in fact right now, as you are experiencing) you are going to be the bad guy.
Why is that? Well first of all I'm guessing you left him. For years you probably had the alcoholic in the house. People walking on eggshells, making sure he doesn't hurt himself - Dad has a problem. The alcoholic/alcoholism is the center of the house, the drama, the cleaning up after.
Then YOU decide to stop it. You leave or ask him to leave and now what? YOU have abandoned the very person you have all been looking after all these years.
That's the start.
I did read in your posts that you have an agreement not to speak badly of the other parent. Fair enough, I actually agree with that.
What you seem to be doing though is confusing that with the facts. The fact is he doesn't pay child support the fact is he did all the things he did (and I'm sure there were a lot of things that lead to the divorce).
The truth is he is an addict and your Sons are becoming his co-dependents (from what you have described).
You can be the martyr here or you can tell them the truth. Tell them you can't afford to do all these fun things because Dad refuses to pay child support. Tell it in a matter of fact way, write it out, put it on a spread sheet.
Explain alcoholism and addiction to them (in an age appropriate manner). Explain codependency to them. Explain that they are not on this earth to care for their Father. That they are independent people.
If they are in therapy, perhaps this is something you could do in a meeting with their therapist (or find a good one). Think of it as a codependent intervention?
Buy them a copy of Codependent no more by Melody Beattie and make it required reading. Discuss it.
Anyway, that's just my opinion. If you step in now there is hope that this will all turn around.