I have a friend who has adopted two children. Both children were adopted through adoption agencies. For the reasons you listed in your post, I would recommend going through an adoption agency.
My friend's first adoption was a quick and easy process. They adopted a newborn baby girl. When they moved from the East Coast to the Midwest, they decided to use the same adoption agency for their second child. The Midwest branch did not have very many mothers who wanted to give up their children for adoption, and the office didn't seem well run. My friend and her husband languished on the waiting list for more than a year before they gave up and signed on with another adoption agency. The second adoption agency was more expensive, but it was more reputable. Within a month of signing on, they were chosen by a pregnant teenager to adopt her son. Things went smoothly with this adoption too.
The huge benefit to using the second adoption agency was that the adopted son was on the West Coast. The laws were different out there, and the adoption agency helped my friends navigate the different state laws. They also provided counseling to the birth mother to make sure she was comfortable with her decision to put her son up for adoption. In a weird twist, my friend's husband unexpectedly developed feelings for the birth mother. He could not understand the inappropriateness of his relationship with her, and the counselors at the adoption agency helped my friend, her husband, and the birth mother navigate this awkward (but surprisingly common) territory.
I wish you the best of luck. I'm so sorry to hear about your stillborn children. You have three sweet angels watching over you. I hope everything works out with the adoption.