I don't think you need to try to explain or prove anything to your new coworkers. Prove yourself by doing good work.
Slightly off-topic but there were some other things in your post that stood out to me. First, you keep saying your MIL hates you. Instead, why aren't you referring to her as the MIL whom you hurt? It feels like there is a subtle shift in blame or a slight victim mentality you have by repeatedly stating MIL hates you.
MIL is hurting, she is upset and worried about her son and grandchild. Saying that she hates you makes it sound like she is this evil person who enjoys holding grudges. It is a subtle phrasing and maybe I am reading too much into it. If so please disregard. I know you are owning the fallout from your actions. Just don't forget to show others some grace and mercy as they work on rebuilding their trust and love towards you.
The second thing is how you are hoping to start over but that may not happen if word gets out at your new job about your affair. I just wanted to share my experience with this as word DID get out to all of my family and friends and started floating around our small town.
Starting over for me wasn't just about doing new things, new jobs, getting away from AP, etc. It was about changing who I was on the inside. It was/is about becoming an honest and trustworthy person to myself and to others.
So, once word got out I felt very conflicted on how to handle it. I didn't want to fuel the gossip. But I didn't want to lie. What kind of a fresh start is it if you are lying about your past or hiding things? So when one of my close friends called and laughingly told me she heard this funny rumor about me and OM..... well, I had two options. I could have laughed it off and been snarky about ridiculous small town gossip. Anyone that knows me would have a hard time believing I would ever have an affair. Instead, I chose to be as honest as I could without encouraging gossip. I told my close friend the whole truth.
Bringing things to the light, rather than hiding them in the dark, is so much a better option for me now. My BH and I came to an agreement on how to handle it since he had the same concerns about gossip. Now, if I am ever asked about it I plan to say that I did get too close to OM and that it was very inappropriate and extremely hurtful to my BH and my family. Without giving specifics I can still own my actions, admit my faults to others, and use it as an opportunity to reiterate where my heart lies now.