SSN Mistake on Workers Compensation 2-part dilemma

I worked for a major company for two years, and then was fired. My mom signed me up for unemployment compensation, but accidentally used the wrong social security number–it was the social security number of my deceased brother. She mistook his SSN for my SSN, and filed out legal papers and everything. She couldn’t figure out why they weren’t taking my number. She gave me the number to log-in Online with after she’d filed a few claims for me. Now I’ve been receiving workers compensation, but under the wrong SSN.

The second part of my dilemma is that soon after being fired, I got married and moved to California. The unemployment benefits I am receiving are for Washington, D.C. where I previously worked. Soon, I then tried getting a job here in California. I got hired at a company, but they only employed me for thirty days. During that time I received no benefits from D.C. After the 30 days when the California company decided they weren’t going to hire me permanently, I didn’t know what to do. I then reinstated the D.C. unemployment benefits after the 30 days because I still didn’t have a job. Is that illegal? What should I do? Should I immediately inform D.C. about my SSN? I am innocent and scared I’m doing something wrong. I want to right my wrongs. Is it still legal for me to be receiving workers compensation?