My main question about Nick is why he's dead-set on not taking an appeal.
By "appeal" I think you meant plea deal.
In my opinion, Nick has suffered massive cognitive decline over the last couple years likely due to drinking, heavy cocaine use and intentional malnutrition. He rarely has the ability to think rationally anymore. His reasoning about his case, the law and the situation he is in....is batshit crazy. And NOBODY is going to tell him otherwise.
He has thought since shortly after he was arrested that he could beat the charges. He is full of nonsense theories about the law and it doesn't seem as if his attorney is willing to challenge those legal theories or presumably Nick's legal strategy for the case.
Nick is also somewhat delusional about the basic facts of his situation. He has often talked of himself as the victim of a conspiracy and accused just about everyone involved in the case of lying. In particular police officers and people at child protective services.
The other thing that seems to be driving Nick is the idea that he can gaslight away all the facts of the situation if he can just beat the case. That if he wins the case, he can confidently deny that he was ever a swinger, that he ever used drugs, that he was ever addicted to drugs (or alcohol), and that he ever neglected his children.
Lying for Nick is like breathing. And like a small child, he thinks he can get away with any lie if he just repeats it enough.
If he is eventually cornered with no options left, odds are good that he will take a plea deal. Something similar already happened with Child Protective Services. He was fighting them at every single point for a long while, but more recently he seems to have surrendered and is doing the things they require of him.