If I am a non-custodial father, will I still pay child support if I relinquished my rights to my son?

In Arizona, at least, the termination of the rights of a biological parent does not end the obligation to pay support for that child. The obligation only ends when there is an adoption of the child, presumably by a step-parent.

Unfortunately you cannot give up your obligation to pay child support, only your rights to custody/visitation if you so choose. If that was the only reason you were considering relinquishing, then now that you know you would not get the outcome you want child support wise, you should strongly reconsider how you want to proceed, for your son's sake who may later in life want to know his dad, etc. And of course, if you want to pursue and protect your custody rights and the mom is in fact being difficult and unfair, you have every right to go to court on that. And they even have do it yourself forms (at least to get started), but always advised to have those reviewed by an attorney before you file and to eventually save up funds to get an attorney at the actual hearing date.

You can't relinquish unless there is someone who wants to adopt the child (i.e. a step father) and yes, everything has to go through the court because anything else is not valid before the law. And yes you have to pay child support. So it's time to act like an adult, go to court to establish whatever time you want with the child and to establish proper child support if that hasn't already been done.

The court will not allow you to relinquish your rights to get out of paying child support. Talk with a local attorney. I would believe that there is somebody who will take payments. And if she is ignoring a court order, you may be able to get a contempt charge against her and force her to pay your attorney's fees to get it.

Whether the court lets you relinquish your rights will depend on many factors, not the least of which is why you want to do it. If you are successful you will not have any future child support. However, the court is pretty strict on the fact that a person cannot relinquish rights to avoid paying child support.