Same thing happened to me. I have two last names. USCIS based it on what was on my passport. So my EAD came with two last names. The DMV and SSA offices both said they can only go by what it is written on my EAD. Then I went to my home country's consulate (Brazil) to see if they can change it on my passport. They told me its virtually impossible to get my name changed on my passport without ever going back to my home country. Maybe your home country is a little nicer than mine. But if you can get it changed on your passport, then you can eventually get it changed on your next EAD, and then only then will the SSA and DMV change it.

P.S. Of course I'm basing my DMV experience in my home state. Your state may be different.