Hi @Jamie_Long , welcome to the forum 
This may have fixed your issue because the dropdown, to select the Team to move your project to, will show a list of teams that you recently interacted with (or starred, in your case).
What you (and many others) may have not realised is that you can type in the box to search for Teams that aren’t listed in the dropdown. Did you try that?
And also note that the project may still show up in the source team, as well as the destination team, depending on the sharing permissions and members of the project. See no.3 in this post.