I checked with my biography of JA. There's an account about the adoption of Edward from JA's brother Henry, that was told to his niece Anna Lefroy.

Apparently Mr.and Mrs. Knight visited the Austens on their wedding tour and decided to take Edward with them for the rest of the tour, and brought him back after a few weeks. They apparently enjoyed his company and after a few years of married life, it became clear that they were fated to remain childless, they decided to adopt Edward and make him their heir.

In Henry's account Edward stayed with the Knights throughout the following years on and off, but by degrees it came to be understood in the family that Edward was selected as the adopted son and heir of Mr. Knight, and in the further course of time was taken more entire possession of and sent to study in some German university.

According to this biography, the Knights lived in Godmersham, and were visited by the Austens quite often. Later, after JA's father died, Edward arranged for JA and her sister and mother to live at Chawton cottage, which was situated in Chawton park, and Edward's house, Chawton great house, was nearby. Edward's wife Elizabeth had just died and he wished to bind his mother and sisters more closely to himself.