Welcome, asrhodes, I hope we are helpful!
Remember there is no "magic bullet" and you must adjust your program as you see developments.
In reviewing your requests and steps so far, let me offer this program:
For hard clay soil, the actions you have taken so far are beneficial but there is much more you can do to help. As Howard recommends and everyone involved in this forum (so far as I am aware) the "punch and sand" method used on golf courses will in fact cause you more problems in the long run.
If you have not yet fertilized, use a good organic fertilizer with alfalfa in it to stimulate root growth in your turf. When you apply the fertilizer, mix in horticultural cornmeal (or any plain ground cornmeal if you can't get horticultural cornmeal) at about a 2/3 fertilizer to 1/3 cornmeal ratio. Water it in with a mixture of molasses at 2 oz. per gallon of water with the bio-innoculant product Earthworm by Ag-Way. Most of the independent organics suppliers have it or you can order it on line from several places.
If you have already fertilized, apply just the cornmeal, Earthworm and molasses. They will provide your soil with organic material that encourages beneficial fungus growth (cornmeal), innoculates it with a concentration of the most beneficial microbes so you have a good population to do the important work (Earthworm), and encourages the health and vitality of microbes in the soil (molasses).
When you mow, mulch the clippings and leave them on the yard; they are extra organic material that will return nutrients to your soil and aid in the development of better tilth. Do not mow the grass too low; it will consistently stress the turf. Find out what type of grass it is so you can mow it the correct height. That is very important.
After a couple of weeks following the application of these products, slow down your watering program to water more deeply and less often to encourage the roots to grow deeper into the soil, which by then should have loosed up enough for them to gain a foothold. Of course, you will have to adjust the watering so that it does not run off, as you say. You should see a decrease in that problem as the weeks go by.
After a couple of months, an application of Medina Plus would be terrific. It contains microbe stimulants and seaweed which will further encourage root growth in your turf. It might be overkill to use but when you apply fertilizer in the summer, watering it in with Medina Plus will give you a one-two punch that is hard to beat. You should have a well established colony of microbes and fungus in your soil by then that can take over.
Dchall (Dave) how does that sound to you? I believe it should work well. That's what I would do in this situation to my own lawn.
Cheers and best wishes! Take heart!