This will be different from state to state, but in case anyone is looking for how to do it in Pennsylvania:
Step 1 - Registration of foreign adoption (equivalent to readoption in some other states)
You do not need an attorney to do this. Take all your paperwork (Ethiopian court order with official translation, embassy-issued "birth" certificate, Ethiopian passports showing IR3 visa, Certificate of Citizenship, any paperwork that has children's pre-adoption full names etc.) and hand over the ORIGINALS - gulp! - to the Orphans' Court. This is also the time to request a legal name change. In a couple of weeks you will get a call to go pick up the official registration of foreign adoption and get your paperwork back (you can use the mail, but remember, these are your original documents). At the same time the Orphans' Court will forward the information to the Division of Vital Records.
Step 2 - U.S. birth certificate
This time you can use copies, so it's easy to do it by mail. Follow the steps here and send your request to the Division of Vital Records (Ignore the broken link at the top because you've already completed the registration of foreign adoption). For "child's name at birth" put the name listed on the U.S. embassy-issued birth certificate. You will receive a U.S. birth certificates in the mail a few weeks later. Be aware that special characters like apostrophes do not show up on the birth certificate.
Step 3 - Social security card
Go to your local Social Security office. You'll be using your originals again, so better not to mail them. In addition to all the documents listed here, I was asked to provide either an immunization record or a record of a well-child visit. Once I had all the necessary paperwork, I could wait in the office for the request to be put in and take my originals home. The cards came in the mail just a couple of days later.