in such poor soil diversity will help, if you aren't too picky about uniformity you can include other plants in the mix, but late spring is a tough time to get almost any plants established coming into the hot summer months.
having those weeds there providing free fertilizer (when mowed back) is ok. just keep doing what you are doing and then this early fall you can scatter your preferred grass and other seeds to fill in. adding some compost in a thin layer with those seeds will likely help even more.
i like clovers, yarrows, the wild trefoil (which is also a nitrogen fixer) and any other local species i can get going in the mix. i figure the wind, rains, etc. are going to be bringing any weeds in anyways so why fight them? i like dandelions and plantains too.
the main thing is to just keep mowing it and with a sandy soil i would mow a bit higher to keep it more covered and shaded. regular mowing will select the weeds and grasses that can survive. regular mowing provides the uniform look and that's ok with me. weed killers and fertilizers are wastes of $. i'd rather just keep it mowed and that is good enough. the only time i've watered here is when i've been trying to get some new seeds going. that was only few days in some 22+yrs...
if you want to provide more green stuff to the soil for planting into later in the summer and early fall you can plant a cover crop while there are still some chances at rains and before it gets too hot. i'm not sure how buckwheat grows down that far south but it makes a pretty good nursery crop for starting other longer to get going seeds like alfalfa, clovers and the wild trefoil.
ALSO, very important!!!
CHECK with your local land grant university EXTENSION office to see what types of grasses do well in YOUR climate.
Even thought the current trend might be adding clover ( you may have heard this), clover may do well in the wet east, but not in YOUR back yard. They can also identify this weed and give you advice as to deal with it.