"As far as I know " if you have H1b for an employer and if you want to work for another employer under H1B, the future employer must have at the very least filed for a H1B.
Also since you have an EAD (assuming you have filed 485), I think if you change jobs, you are doing so with your EAD.
I am curious about your lawyer who says you can continue working with the same employer under the previous H1B. I dont want to say anything definitely but I think your lawyer is wrong. But if your lawyer means that it might be safe for your new employer to file your H1b then in that case, I dont know what AC21 allows.
Does AC21 require people to change jobs on EAD or can they do so with H1?