tl;dr: lied to a co-worker in an email, changing the contents of previous emails. Looking to find a path to regaining their trust. How do I continue to work in close proximity, whilst there is this uncomfortable tension?

A senior colleague asked me if I had ignored the software release procedures, in a recent project I had worked on. The procedures related to the deployment of our software products for internal clients. I initially told him that I had followed procedure, but after reviewing my old emails, I realized that I had sent out uncontrolled/untested prerelease versions of the software multiple times. This is not an uncommon occurrence at the company, but given my role, I should have known better. My initial reaction was one of embarrassment; how could I fail to remember something so recent.

I wanted to be forthcoming about my mistake, but I was still feeling shame for having been so confident in my incorrect answer. I sent him an email, showing him the last instance in which I had failed to follow the release procedure, but I choose to redact all of the previous instances from the email chain. I realize that this is equivalent to lying.

It was not long before they had realized what I had done. A couple of hours later, my manager books a meeting with me, regarding that project. At this point, I was blissfully unaware. When the question arose, I confessed and to his credit, he was kind and understanding. I justified my not following release procedures by stating it was necessary to get quick feedback on feature development, and that the users were aware it was pre-release. Needless to say, I didn't feel too great for the remainder of the day. My colleagues, who had informed my manager, choose to ignore me for the remaining hours of that day.

I realize the correct course of action is for me to formally apologies to my colleagues. I plan to do so. I also realize that I am entirely in the wrong here. However, even in the best scenario, I cannot see a path to regaining their trust. I do not deal with conflict/tense situations well, so continuing to work in close proximity will be challenging, especially since that project is ongoing.