When you say that new sod was just laid in your yard, was it overseeded with winter ryegrass when laid this past winter or was it recently sodded as bermudagrass only in the past month or less?
If there is overseeded ryegrass, it should be scalped out or mowed at a shorter height of cut now so the bermudagrass emerges from winter dormancy and does not have to compete with the rye. Maintain watering at a bit less than was required for the ryegrass but enough to keep the roots of the bermudagrass moist since it was use to getting watered all winter. Follow irrigation scheduling based on AZMET https://cales.arizona.edu/azmet/az-turf.htm
Now is ideal to fertilize the lawn with nitrogen. Your Scotts product has 11% N from organic sources like bone meal, etc. You would need to apply about 4.5 lb of the product per 1000 sq ft. Comparatively, you can apply less expensive ammonium sulfate (21% N) at 2.5 lb per 1000 sq ft now. Ammonium sulfate dissolves quickly when irrigated immediately after application. Using about 0.5 lb of N each month should green up your lawn for the summer. Reduce the fertilizing before overseeding in October.
In fall, a complete fertilizer with phosphorus (middle number of fertilizer analysis on label, N is first number) will aid in getting roots established for the overseeded ryegrass.
If you have a healthy and vigorous lawn, you should not have to spray preemergence herbicides for weed infestations. A new lawn could be picked by hand if only few weeds appear. Corn gluten meal is not safe to use on turf because it will control both grasses and broadleaved weeds.
Bermudagrasses require lots of light and suffer under trees in shaded conditions. Try to lift up tree branches so more light can penetrate to provide light. Raising the mowing height of cut will enable the grass to grow taller to be able to have more leaf surface to capture light as well.