(a) Harris says Trump is unethical and cites the example of Trump gate-crashing a charity event to falsely get credit for himself. Adams responds by saying that others are equally bad—that all politicians do morally dubious things. When Harris points out that Obama would never do such a thing, Adams says Trump is a very public figure and hence people have lots of dirt on him.
There's nothing wrong with Adams here, because
1) Harris's implicit argument is that Trump is unethical compared to other politicians, even if he doesn't actually say it. Thus, pointing out that other politicians are unethical is a legitimate rebuttal.
2) As far as I can tell from your summary, the argument is not "Trump gets caught at more bad things than other politicians", the argument is "Trump does more bad things than other politicians". If someone brings up an example of another politician who doesn't get caught as much, it's entirely proper to point out that it's harder for that politician to get caught and that not getting caught doesn't mean not doing bad things.
Instead, Adams always says, Trump “doesn’t pass the fact checks”. This move essentially makes it sound as if there’s some organization whose arbitrary and biased standards are what Trump doesn’t pass and so downplays the much more important fact that Trump lies.
This has some of the same problems as #1. Pretty much nobody tells the truth 100% of the time; saying that a politician lies really means he lies more than other politicians. Just because nobody explicitly said "... more than other politicians" doesn't mean the implication is not there. It is entirely correct to rebut this by saying "actually, other politicians lie, they just don't get caught at it by fact checkers".
Furthermore, if you don't think there are organizations that use arbitrary and biased standards for Trump lies, you haven't been paying attention to the controversy over fact-checking sites. Generally those sites are correct when it comes to facts, but selective about which facts to check and whether a literally true or false statement counts as mostly false or mostly true.
Recently, James_Miller posted a conversation between Sam Harris and Scott Adams about Donald Trump. James_Miller titled it "a model rationalist disagreement". While I agree that the tone in which the conversation was conducted was helpful, I think Scott Adams is a top practitioner of the Dark Arts. Indeed, he often prides himself on his persuasion ability. To me, he is very far from a model for a rationalist, and he is the kind of figure we rationalists should know how to fight against.
Here are some techniques that Adams uses:
Overall, I think what Adams is doing is wrong. He is an ethical and epistemological relativist: he does not seem to believe in truth or in morality. At the very least, he does not care about what is true and false and what is right and wrong. He exploits his relativism to push his agenda, which is blindingly clear: support Trump.
(Note: I wanted to work on this essay more carefully, and find out all the different ways in which Adams subverts the truth and sound reasoning. I also wanted to cite more clearly the problematic passages from the conversations. But I don't have the time. So I relied on memory and highlighted the Dark Arts moves that struck me immediately. So please, contribute in the comments with your own observations about the Dark Arts involved here.)