What grass are you growing?
Most weeds can't handle irrigation and organic fertilization. Bermuda and St. Augustine like that treatment so they crowd out the weeds. A few exceptions include dalis grass. This is why the weeds are so happy. You have made it worse for the grass.
Ultimately you need to follow organic guidelines to support the grass growth. CGM is a good start because even if it fails to wipe out the weeds it is still a very good fertilizer.
Marshall Grain Company is a reasonably priced large supplier of the products you will need. Their website is www.marshallgrain.com
Other great nurseries are on Davis Blvd. Green Mama's, 5324 & NAtural Living, 6616.
Good luck with your project. Let us know how it goes and ask more questions.