Here's some information about
Adoption (firsthand as an adoptive parent and also having working with foster and adoptive families). Laws seem to change continuously from state to state but right now, same-sex couples in all states except Mississippi adopt.
https://www.lifelongadoptions.com/lgbt-adoption-resources/lgbt-adoption-laws
If the laws prohibited the Hart's from adopting as a couple, there is something called "A second parent adoption." A second-parent adoption allows a second parent to adopt a child without the "first parent" losing any parental rights. In this way, the child comes to have two legal parents. In other words, after one partner adopted the children, the partner could then petition to adopt the children. My guess is that both parents adopted the children.
https://www.childwelfare.gov/topics/adoption/adoptive/family-type/second-parent/
Regarding foster care and adoption and monthly stipends. Stipends are provided to families who provide foster care. The stipends vary from agency to agency and from state to state. Some sites will tell you what the stipend is if you google it. Stipends are typically $550- $1000/ per month per child. There are different levels of care as well and special needs children would typically provide a higher stipend. The idea behind the stipend is to reimburse a parent(s) for food, clothing, entertainment, utilities, etc. I would like to believe that families go into foster care because they are passionate about children and believe that these kids, typically with a trauma history, need an advocate.
When a foster family makes a decision to adopt a child, that stipend will continue to be provided to the family if it is needed/ wanted until that child leaves the home or turns 18. Stipends will continue even in the case of kinship care, e.g.,, a grandmother adopting a grandchild. The child will also be provided with Medicaid (healthcare). Without the stipends, fostercare would be in an even greater mess than it is right now. There are fabulous foster and adoptive homes out here and as we all know . . . there are homes that should never be permitted to have children.
One way of adopting children is to become a foster parent in your county. Bio parents have 2 years (Adoption and Safe Families Act) to "fix" reasons why kids came into foster care. After that (and that too varies) the goal is supposed to be legally changed from reunification. . . to adoption. Another way to adopt is to look at available children through SWAN, (Statewide Adoption and Permanency Network) whose parents rights are already terminated, and that may be going to another state. It looks like that's what SH and JH did.