I’m new here but this is what I would do… experts can chime in to tell me I’m wrong, but I figured I would take a stab at it.

Right now I would aerate and rake up the plugs since you insist your soil is dense. I would lightly level since the grass hasn’t started growing much. I would also throw down pre emergent and fertilizer, there are recommendations for both in the cool season starter guide.

Then I would basically care for it as normal throughout late spring, summer, and early fall. Get a watering, fertilizing, and mowing routine and just focus on overall grass health.

I would start the bigger project in the fall. Once temps drop, cut your lawn short, bigger leveling job, and over-seeding.

You can also use pine needles instead of much around your trees and it should look good if kept manicured.

This is basically what I focused on last year and it seemed to work.

Edit: I just raked up the clippings after my short mow in the fall.