If you used synthetic lawn fertilizers then that is not organic. If you regularly use synthetic fertilizers on your lawn, that does not make your soil 'rich' or 'fertile'. Your lawn may be lush because you apply alot of synthetics. It could be weed free because you use synthetic weed killers. Synthetic fertilizers feed the plants, not the soil. Organics feed the soil that feeds the plants. Since you don't say what fertilizer you used, it's difficult for me to start with the organic, rich or fertile idea.
You say you had your soil tested and have a pH level, but you don't say if you had nutrient contend tested, so I'm at a loss here. The best way to make your lawn, flowers and veggies organic is to add organic material to the soil, not use synthetic fertilizers, synthetic pesticides or synthetic herbicides. The best way to make the soil more organic is to add compost. You can make your own or purchase it bagged or in bulk. Here's a compost calculator.
Adding compost to a raised bed will help the soil to retain moisture and improve the tilth (texture) of the soil as well as the good microbes. If your soil is sandy it will drain quickly and dry out more quickly. The smaller the area of the raised bed, the more quickly it will dry out. If you have the space so you can garden from either side of the raised bed, make it 4' wide and as long as you like. At 4' wide you will be able to reach into the center from either side. For a raised garden bed, I like to use 60% screened topsoil and 40% compost mixted together.
Maybe this will be helpful. The site about gardening organically is from Florida.
Here's how to maintain your lawn organically.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.