This is such a hard thing to wrap my head around, largely because of all of the shared history. That still happened. What happens to that shared history moving forward with what you describe as a new marriage? What does that feel like to you now?
For me, I first had to get rid of a lot of "black and white" thinking. There are no "good" people and "bad" people. We are all capable of being both in different areas of our lives. For a moment, imagine someone you might know, a pillar of the community, who volunteers their time, donates to charity, rescues dogs, etc. Then imagine turning on the TV one day and seeing that same person being arrested for distributing child porn. What do you think now... is this a good person or a bad person? Because the good things they did were still good, even though they did a bad thing. And the bad thing they did is still really bad, even though they also did good things. Lastly, is the fact that they did those good things in any way, shape or form, a reason to excuse, justify, minimize or forget about the bad thing they did?
The truth is, we are what we do. During the affair, my days were filled with the need to lie, and hide, and deceive. I felt like a bad person, and justly so, because I was being a bad person. One of the bigger mistakes I made along the way was to start identifying myself as a bad person. If you are a "bad person", as opposed to "a person", then you have no hope, no path forward, because it is who you are at your core. But a person, by contrast, has choices, and makes choices based on their desired outcomes. If you no longer want to be a liar and cheater, then be an honest and trustworthy person. There is no one and nothing stopping you from never lying again, never cheating again. Everyday, I wake up, and ask myself who I want to be. And to some degree, who I do not want to be. And so I choose to tell the truth, always, and right away, even if it throws me under my own bus. That allows me to sleep at night, because I do not bear the burden of hiding lies. Honesty is like Ambien :) I choose to do something nice for someone else every day, and I do my best to do things that have no benefit to me (because it's important that we don't do nice things in order to manipulate others into loving us), such as holding a door open for someone, letting someone cut in ahead of me on the highway, allowing that person with one item to get ahead of me in line, or even just telling someone they have a nice smile. I do my best to live a life of integrity, honesty and dignity. Because that is how I live my life, that is how I see myself, and it how others tend to see me as well. That, my friend, is how you get out of the shame spiral.
So... back to your question. What does all that have to do with moving forward in your relationship? Well, the two have a lot in common. What I found to be super helpful to me was to view both my life, and our marriage, as a story. Stories have chapters. The heroine may be tempted to do the wrong thing in chapter 3, but by chapter 7, she's turned her life around. Our lives are the same. Your affair was chapter 3 of your life. You did the wrong thing, and sadly, you can't undo what's been done. While we cannot control the past at all, we have a lot of influence regarding the future. So, simply accept that Chapter 3 is only PART of your story. Now you are in a new chapter, and the other chapters haven't been written yet, so you get to write them.
Your marriage is the same. You were both in a different place before the A, during the A, and now after the A, and each of these is a different chapter. Undestand that chapter 3, "the marriage you thought you had", is over. So now, both and your spouse, get to write the new story together. Again, for me, it helped to think of this as starting from zero, as if we had never met, and you build a new relationship together, one based on what you now know about yourselves and what you want and don't want. No more "blind faith". No unconditional love. Rather, faith and love are something you build through openness, honesty, and empathy. You prove yourself as someone they CAN love, just as you did when you first get married. Just a little more based in reality this time. How this happens is up to you. But start by establishing who you both are and what you both want from the relationship. What does a good marriage moving forward look like to you? How do you ensure that communication flows? You two write your own story together moving forward. If there are things from previous chapters that still hold value to you, please feel free to bring them into the new relationship, and leave the things that didn't work behind.
Good luck to you. This is along, hard road, but the rewards are worth it.