Can you finish sweep the board in june's journey without using diamonds?

Yes, I have never spent diamonds, and I've played them all except 3, when I didn't have the time. You have to figure out what scenes you have (everyone gets different scenes) then go to the game and practice them. Or if it's a scene you haven't reached, study them. There are many resources, such as GamersUnite or YouTube If you need help finding a scene, the best is to join https://www.facebook.com/groups/sweeptheboardoriginal where they have dedicated players for quick assistance and all the resources and info. Also helpful to have many tickets, so you won't be tempted to just spend diamonds. Then just consider it practice and try again.