Adoption through foster care is free or very low cost. Children available for adoption though are generally over 3 years old but it is possible to have a toddler/infant/newborn placed with you too. My daughter came to me as a temporary foster placement as a newborn. I picked her up from the hospital after she was born and adopted her a year later. She's 17 now.
Infant newborn private adoption through an agency or adoption attorney can cost $20,000+. Specifying you only want a girl can make your wait time longer. Some agencies won't take you if you're only open to certain types of situations.
Most birth parents pick out the adoptive parents these days and from profiles submitted to them. Many adoptions now are "open" adoptions meaning the birth parents are a part of the child's life growing up. Adopted kids quite benefit from this type of arrangement. It can be anything from visits a few times a year, picture updates, video calls, etc. Though adopted from foster care, my daughter has this. Her bio mom died when she was a toddler but my daughter has always known she's adopted and has frequent contact with her biological family. She sees a grandma, aunt and a half sister she has that is being raised by her dad (the sisters dad) and they are all part of our family too.
Another thing that might make you wait longer is that you have biological children. A lot of birth parents choosing to place their baby for adoption, often choose a childless couple who might have fertility issues, but that's not always the case.
Every adoption type in the US requires a home study and a home study for a private adoption is $1000-3000. An adoption home study through foster care is free. Some agencies and adoption attorneys do accept home studies done through foster care, but most do not. A home study consists of a criminal background check, fingerprinting, financial disclosures, 3-5 references, interviews with you, interviews with your twins (age appropriate questions), a physical from a physician, parenting classes (including CPR) and a physical inspection of your home. None of this is optional. My husband is a Cardiologist and even he was required to take CPR classes for whatever reason.
In a private adoption, the adopting parents are responsible for the birth parents legal expenses. Some states allow you both to use the same attorney, but some also require separate legal representation for both parties. You may also be responsible for some of the birth expenses and prenatal expenses. That can be anywhere from help with rent, doctor copays, help with groceries, etc. Most states have a limit of how much adoptive parents can pay for this.
TPR - After the birth and the birth parents sign TPR (termination of parental rights) there is a revocation period where they can reclaim the child. Revocation is anywhere from 3-30 days in most states but some states it's all the way up to adoption finalization which in most states is about 6 months. If they choose revocation and reclaim the baby, their parental rights are reinstated and your expenses up to that point are lost (you're not refunded anything you've paid). In a foster care adoption the parents rights have already been terminated, usually by the state.
Once your home study is approved, you're not limited to adopting only in Indiana, you can adopt from almost any state (for a foster home study, you can adopt from almost any states foster care system). Limiting to only Indiana, you will likely wait longer. If you do adopt from another state, you can't leave the state with the baby until you get clearance from a judge to do so which can be 7-14 days (sometimes longer) which is known as ICPC (Interstate Compact on the Placement of Children).
If you are able to self match with expectant birth parents, this can dramatically reduce your expenses. This is generally someone you know or you get in contact with through other means (friend of a friend as an example). This changes the adoption type to a direct placement and then you're really only paying legal expenses which can be a few thousand dollars. Direct placement attorneys often do accept a home study from foster care. My MIL is an OB/GYN and she has helped some patients considering adoption for their baby direct match.
Even though you're not planning to adopt for a few years, since your wait can be a few years you might actually want to start the process now.