Yes, Shape Builder should work fine.
- Select both the overlapping shapes.
- Select Shape Builder tool (Shift + M).
- Now, you have to remove those shapes using this tool. To remove a
shape, you have to hold Alt (Alt on Windows, Option key on Mac) key. When you hold it, it will show - (minus) sign, which means you're going to remove it. By default,
it shows + sign.
- Click the areas which you want to remove.
Alternatively, you can use Divide feature in Pathfinder panel. It will basically split it into different shapes. Once applied, it will split and group everything. You can then delete the shapes you don't need from the group.
Note: This method will remove overlapping effects. The overlapping parts will be gone. It will be replaced by a new common area. Basically, you'll get 3 different shapes which are adjacent to each other.