I am currently going through a domestic case with my recent ex-wife. Just the other day my public pretender, and that's not a spelling error, called me outside of the courtroom and asked me to plead guilty to all of the offenses. I then told him that I would plead guilty to what I have done, which is I violated a protection order when her brother told me she wanted to talk about Ollie our daughter. When he told me that I had prove my innocence, I interrupted him with no I don't, our Constitution says I'm innocent until proven guilty. He then told me that the Constitution doesn't apply to this. Now this States Attorney has declared me a danger to the community, with absolutely no violent crimes in my record, I have 2 DUI, and a federal prison sentence for conspiracy, which I believe the way I was indicted is now illegal
The courts do not supersede your constitutional rights, although you may believe that you have a constitutional right that isn't actually there. This article discusses the position that "due process forbids convicting an individual of a crime unless the government proves the elements of the charged offense beyond a reasonable doubt". This standard is actually not stated anywhere in the US Constitution, but it has been assumed as an implicit meaning of "due process".
It sounds like you were charged with a crime, and there is most likely an applicable statute in your state that is analogous to RCW 26.50.110 in Washington. So you have the right to a trial and the prosecution would have the obligation to prove all of the elements of the crime beyond a reasonable doubt. It also appears that you did violate the applicable law and you were willing to plead guilty, as urged by your attorney. You are correct that you don't technically have to prove your innocence, but there is a practical problem that if the prosecution provides some weak evidence that you violated the law, then the jury might decide that your failure to refute the evidence means that there is no reasonable doubt. The problem is that there is a tendency for jurors to think that the defendant has to create a doubt. States differ somewhat in how they explain the burden of proof to jurors, and you might fare better in a state where the instruction is that "you must be firmly convinced".
Since the attorney seems to have said that "the constitution doesn't apply to this", this is a puzzle. I would not assume (though it is possible) that the attorney was incompetent. It is possible that he was speaking of a non-criminal matter, and it is possible that you were talking at cross purposes. There is no legal situation where "the constitution doesn't apply to this", but perhaps "that constitutional limitation doesn't apply to this specific situation". Regardless of what the attorney said, your attorney doesn't violate your rights, even if he gives you bad advice. The actual court might, and then you would have a cause for an appeal. Similarly, if the district attorney reasonably believes that you are a danger to society and is prosecuting you, that is not a violation of your constitutional rights. An improper conviction would be a violation of those rights, although it might take an appeal to get the court to recognize that fact.