Hi Greg!
Well what I can say is "keep on keeping on!"
Keep mowing high, weakening the weeds growing now.
Stop throwing seed until the fall. Summer won't be nice on your seedlings.
Stop fertilizing until the fall, when you can try again to throw some seed.
Enjoy the "nice and green from afar" aspect of your lawn

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If the soil was decent before, and if what soil you added (if any) was decent, then it will be good again without too much trouble. For now, let the weeds put in their organics and pave the way for what you'll get going in the Fall.
Any instant gratification will be a lot of work, and in my opinion not worth it in the end, since you'll be doing a lot of stuff to sustain the seedlings that I wouldn't want them to grow accustomed to, while at the same time constantly fighting the weeds that will take advantage of the coddling environment and fighting the summer heat.
I think the most instant gratification would be sod, also the most expensive and still worrisome in getting to survive the summer heat.
Next in line would be dividing the lawn into sprinkler-reachable sections, clearing and seeding/watering/coddling/(strawing?) new grass into place, 2-4 weeks per section. Sprinkling much more often than spring or fall to fight the summer heat. Pulling weeds as they come. Whew! makes me tired just think about it!
Next I see waiting until fall and letting mother nature help.
Vinegar may help thin the weed herd, but with nothing to take the place of the weeds (unless you baby it intensely) I think it will either give you a worse looking dirt lot, or just let another weed pop up, possibly one that's tougher in resisting vinegar!
I don't want to sound TOO disparaging, I just don't want to futz around working much harder than you need to.
