The reason for the rise of independent adoptions is that birth parents will not go to the traditional agencies. The agencies will not let them pick or even meet the adoptive families or guarantee immediate placement of the babies (often preferring to bounce them around in foster homes for a few months), and they do not permit any medical or financial assistance from the adoptive families, even though such assistance is perfectly legal.
There are non-traditional, nonprofit agencies that refer calls from birth parents to adoptive parents and that let them work things out themselves--you can avoid lawyers entirely. The cost at the agency through which I adopted my two healthy Caucasian newborns was $200 per child, in the form of tax-deductible donations. This agency is listed in the Los Angeles Yellow Pages.