Forget about amending the soil, or using pesticides to kill weeds, Duane.
The best next step for you is to rent a 'sod cutter' from an equipment rental store. Easy peasy to use, cheap and it will cut out all of the soil and weed crowns like now. 2" deep...or 1 1/2 inches. These pieces of sod make valuable soil for plant beds. Just pile them up upside down then cover with 4 inches of soil. Don't send this stuff to the dump.
You need to focus on defining a beautiful edge to your lawn. That is what the eye sees, those edges and as long as your 'crop' is uniform that will look beautiful.
When you make a curve keep that radius going until you decide to change to an outside curve. The radius size doesn't matter as long as you keep it consistent until you change the direction of the curve. Use string, a stake, even a hose works well. Make a 6"X6" trench all along the edge of your lawn. Use a flat shovel. Throw the grass and soil up on your beds. Simply redo your edges in the spring with a shovel and/or use a string trimmer. I'd buy one of these, a Stihl. Always wear eye protection and watch where the debris and rocks fly.
After cutting the sod, grade and rake the newly exposed soil after cutting out the sod. Then roll with a water filled roller also cheap to rent! Fill in any depressions and then roll again. Do this again. That surface when firmed with be the surface of your crop of grass. There should be at least a 1 to 2% slope to carry the excess water off of your lawn and directed somewhere the water should go legally.
Next let's talk about WHY this happened to your lawn. You most certainly have cool season grasses. Always a mixture of species. Make dang sure that the label of lawn grass seed states zero weed seed.
I'd be more inclined to use sod rather than seed. Far better to discourage any weed seed germinating during this time.
The next few suggestions are the most important thing you need to remember so this doesn't happen again; never mow lower than 3"! 3 1/2inches is best. Always sharpened blades. Always changing directions with each mow. Bag your clippings and use them in your compost pile or dumped on top of weeds in the back of your plant beds. Make sure the grass is no shorter than 3" after mowing!
Next critical thing you need to learn is how to water. Never ever every day. Water deeply and do not water again until you see your footprints stay down in the grass.
Next would be fertilizing. Three to four times per season. I use Dr. Earth's Lawn Fertilizer designed for the period of the season; spring, early summer, summer and fall! Dr. Earth's stuff you would only need 3 applications per season.
The next critical thing you need to do once per year is aerating! Pulling plugs of sod and soil out of the lawn then allowing them to stay where they fall. This is another piece of equipment to easily rent. Share this expense with your neighbors and have an annual PARTY!
You should not have to deal with any other chemistry or equipment. Raise the deck on your mower and if your mower can't handle leaving 3" of top growth take it to your new buddies! your lawn equipment service place. They'll be able to retrofit the height of your deck.
Critical height of cool season grass is 3 inches. Any lower there is not enough photosynthetic factories to support the grass plant. Our cool season grass lawn species are genetically gifted with large and deep root systems. They need top growth to support these large plants!
Cool season grasses have to be watered deeply then the soil needs to be allowed to dry out. As the moisture is used and evaporated from the surface the roots grow deeper to reach the moisture deeper. 4 to 6" deep. Watering a little every day is the worst thing one can do for cool season grasses.
Water deeply (remember the time it took for whatever irrigation method you are using). Use a shovel, dig down to see the soil profile. Simply jab the shovel into the lawn and pull back exposing 4" of lawn bed, the soil. When you see water reaching down 4" beneath the surface that is the correct amount of time.
Do not water again until...you walk on your grass and that grass stays down showing your footprints clearly. Perfect time to water and water deeply again. You should be able to train your grass to use only 1" of water per week. And during times of drought? Your grass will always be green!
The proper amount of top growth is able to feed these large root systems via photosynthesis (this is where the fertilizer necessary as photosynthesis doesn't happen without NPK and a good dozen or so microchemicals,(not to forget CO2). We humans are responsible for all of this chemistry. Leaving clippings on the lawn is silly. There is nothing left of chemistry of old plants to pass on to the new grasses or new plants. If the clippings aren't decomposed quickly enough they become thatch which completely ruins a lawn.
I sure hope this helps. There are lots of other question answers about lawns on our site for those who want to understand and grow their lawns efficiently, save water, promote healthy grass crops that are able to resist insect damage and disease. Lawns aren't that expensive or tough to manage if you know the rules!