Can I adopt children before receiving the letter about adoption?
Solution 1:
In addition to adopting from the orphanage in Riften, you can also adopt some children in other cities as well:
- Alesan in Dawnstar
- Blaise in Solitude
- Lucia in Whiterun
- Sofie in Windhelm
See the Elder Scrolls Wiki for more information.
Solution 2:
You need to
- have the Hearthfire DLC
- install the kids bedroom in one of your city houses or
- have one of the houses that you build yourself in Hearthfire
To calm down the kids in the orphanage you can use a calm spell or shout.
See more information here.