This is what I ended up doing. It's a little time consuming, but in the end it got the job done.
- Go the parent/connected project and select the task(s) that you wish to move to a different project
- In More menu, select "Move to," enter the Destination Project. Move task.
- Go to the board that is connected to the "new" project. In my case, the new card landed in my Intake column so I needed to move it to the correct column/status.
- Go back to original board and find the card that was moved (it still shows there, but connected to the new project).
- Open the card, Disconnect from the project.
- Still on the original board, with the card closed, click on the three dots, then Delete (card).
There may be easier ways to accomplish this, but I was hesitant to try other things because we didn't want to risk losing any data.
The moved cards retained descriptions, assignments, comments and attached images. They did not retain the Tag, though.