Misdemeanors committed in the past, when there is a reasonable explanation and rehabilitation is shown, will not keep parents from adopting a child. More serious, and/or more recent criminal charges are issues that may result in a delay of several years or even a bar from ever being able to adopt a child. These will require further discussion with a CC Adoption Coordinator.
In Texas, our background checks for prospective adoptive parents are run by the state and you will work directly with them to complete their process. They will send you and CC an email stating you are eligible and approve or deny your status.