Extreme clayey soil

I have recently been a first time owner of a house which has about 6000 sq ft of Bermuda lawn. This house was built about 3 years back and the Bermuda lawn was set by the builder using sod.
The problem I have is the soil is extremely clayey. Even now after 3 years I can pull the grass out and it come to my hand easily. Lawn would be green during the growing season but it is not dense.

This year I did a core lawn aeration and even after a month the holes are intact, not sure if they will ever fill.
I do plan to do a sand leveling in the summer when the grass growth is more and can emerge out of the sand.

Yesterday I applied pre-emergent and irrigated the lawn and it seems all the water would not go it to the soil. I am not sure if pre-emergent would be able to get into the soil and be able to get the barrier needed to stop the weeds.
I scalped my lawn a week ago more weeds are cropping up now.

Any suggestion are welcome on how to make the pre-emergent more effectively absorbed by the soil.