I also got a new job shortly after separating. It's awkward for sure. I tried to keep my business my own, but right away I was dealing with the hiring paperwork that required that I give out information on my spouse in case of accidental death. It wasn't an option to not put him on the paperwork without getting his signature on it. Given that I wasn't even speaking to him, I wasn't about to ask him to sign something like that. The fact that our not-really-messy divorce took two years to accomplish is a testament to how slowly he has been turning around paperwork.
The next thing that happened was that his lawyer claimed she needed to contact my boss to verify my employment. So, I got to tell my boss and warn him. He was very supportive, but it went against my instincts to separate my work and personal business.
Even worse than these smaller annoyances has been the process of changing my name. My employer's policy is that everyone's email address is their legal name. It doesn't matter that I didn't want to go by that name anymore. So, now that I've finally divorced, and finally legally changed my name, after nearly two years of everyone knowing my old last name... Yup, everyone in the entire department gets an update that my name has changed as well as my email address (causing countless confusion in the process). In response to that notification, I receive many supportive emails congratulating me. They have no idea how appropriate that is because they think I just got married. Definitely, awkward.
Sometimes, I still struggle with the unfairness of it all. But, mostly, I try to realize that this is temporary awkwardness. It will pass, and I will be confident in saying my true name again to anybody needing to know it, and I won't require any complex explanations of anything to make sense of it. It's a transition, which I believe is not always graceful, but you will find yourself on the other side someday, and let out a sigh of relief.
Peace,
Fathom