You are going to need to get a homestudy done. Because of the quick time frame they will probably charge more to get it "rushed" but you should definitely be able to get everything done and be able to take the baby straight from the hospital.
Our situation was a bit different as we were actively trying to adopt. We found our atty in our state and when we were matched, she recommended an atty in bmom's state. However, we had to make all the calls, get it all arranged.
Thanks for your compliment about my dd. She is pretty cute if I do say so myself
Casey