If I understood the question correctly, you want to achieve two layers:
- First that contains only overlapping features
- Second that includes features with no overlaps
Let's assume there is a polygon layer "grid"
with its attribute table accordingly, see image below.

Step 1. Apply the SAGA's module "Polygon self-intersection"

Step 2. Apply the following expression if("ID" LIKE '%|%', 0, 1)
in the Field Calculator.

Step 3. Continue with a "Split vector layer"
using the "key"
-field created at the Step 2. Application of "Fix geometries"
on this step may be necessary. And get the output, see image below.

Note: Mind that the final output(s) is(are) in .gpkg
-format.