I have my first house in Whiterun, and I built a children's room there. I also have Lakeview Manor and built the children's beds and chests there too. Now I'm attempting to adopt, and the only home I can put a child in is the house in Whiterun. I would prefer to use my other home though. Is there a way to change which house they go in?
Well from these two forum posts, one potential solution is to ask your spouse to move and go through that dialogue. It might work as long as you have the prerequisites required for children to move in (bed/chest).
If that doesn't work, it looks like a bug that needs to be hotfixed so you might be out of luck.
It's because when you built the house it didn't add to the number of houses you own. Meaning that, as far as the game goes, the house doesn't even exist. It is a bug. A very, very, VERY, ANNOYING bug. -__- There is no fix for it. You will have to load back before you built or purchased the land, until it shows in your game stats that you own another house. (Viewed from the pause menu) That is, if you see it that important for your spouse and kids to live at Lakeview Manor...
Make sure you completely furnish your houses. The child's bedroom, whether it's the bedroom wing, or the upstairs bedroom with the 2 single beds, they need to have a container. Craft 2 child's beds and chests in the bedroom wing or 2 single beds and one drawer upstairs. Make sure you marry somone if you haven't and tell your spouse to move to Lakeview Manor.