Cori, Yes, you must be LDS to use LDS Family Services to adopt a baby, however, they will happily work with birthmothers from every background, without cost or obligation to the birthmother.
I worked for years as an adoption volunteer, was part of an adoptive parents support group and am an adoptive parent of two children. Yes, the process can be long, yes, it can be expensive and it can and is very emotional. There are several different paths to adoption, some more expensive than others.
Foster to adopt is a wonderful path. You become certified as a foster parent in your state, and the opportunity to adopt children in the foster care system can sometimes happen pretty quickly. Many of these children come from less than ideal circumstances, hence, the reason they are in foster care. Sometimes children have been so abused in one form or another they struggle to bond with foster or adoptive parents. Other times, you get them through the initial struggles that lead them to foster care in the first place, and they become an amazing part of your family. In the case of foster to adopt, there is usually no cost. There will be heart ache and love and joy and a beautiful child.
International adoption can be very expensive, quite time consuming, lots of red tape, and you will nearly always wait so that the child is 9 months or older before you are able to bring them home. There are lots of groups and agencies that facilitate international adoptions, do your research.
Religious agencies, (i.e. LDS Family Services, Lutheran Social Services and Catholic Charities) are wonderful organizations to use for adoptions. If your church has such a program, I would offer than those are the best places to start. The adoptions are usually less expensive than international or private/legal services.
Private Attorney adoptions are probably one of the most expensive routes to adoption, at least in my experience. Now, I will tell you something that I don't know for sure, but again, is my understanding. A private adoption attorney will dot all the I's and cross all the T's, but the birthmother probably won't get any counseling or help in mourning the loss of her baby, which is a vital part of this process. More on that later.
Private Adoption Agencies are plentiful, usually for profit organizations, will generally offer counseling or help to the birthmother, and prices can vary. Again, do your research with these agencies, ask them for names of other families who have adopted and ask the adoptive families what their experience was like.
Ask everyone you know for references and suggestions on local agencies or programs. Your best source of advice is from adoptive parents, or those who work in the social services industry.
As for the amount of contact you will have with the birthmother, it is a decision that is made by the birthmother. Those decisions will vary, depending on how much contact she wants when the placement process is over. One of our birthmothers had initially requested we be there when the baby was born. At the time, it was not possible to do that for a variety of reasons, and we did not see our son until the day he was placed with us. We look back on that and know it was the very best thing in our case. Our birthmothers needed time to be with their baby, to bond a little with them and say goodbye. It's very important for that process to take place between a birthmother and her new baby. It is very hard, and yes, some birthmothers will change their minds when the baby is born, but for her to be able to mourn her loss (which is real), and move on in her life, that process must take place.
Find a program that works for you. Discuss with your husband what your wants and expectations are before hand so that you can have an honest disucssion with an agency representative and then a birth mother. Adoption has changed over the years, and it is very common now to find that open adoption is the "norm". Our oldest just turned 12, even in the last 12 years the process and relationship with adoptive parents and birthmothers is completely different than when we brought our first baby home. We love our birthmothers, and what little time we were able to spend with them prior to the birth of our boys is precious to us.
Good luck in your adoption process!
G.