I don't want you to feel ignored, however, you've got more than one problem going on. I'd suggest checking with the Texas forum for general information on growing conditions. Really great people over there!

And check on the Lawn forum for how-to's on improving the lawn. Try to be specific about those weeds! There are lots of pictures on the net, and GW even has a Weed forum which can help you.

You won't find too many people advising use of any commercial weedkiller in conjunction with having a safe area for your baby. Most commercial herbicides just aren't safe. There have been lots of debates here and elsewhere, but in short, the man-made chemicals eventually make way into the Earth's waters which means they eventually [sooner or later and in one or another form] are in the foods you eat. Especially in the Lawn Forum, you will find reference to CMG (cornmeal gluten) as a weedkiller -- this IS safe for use on your lawn, and is extremely effective.

Congrats on the cypress vine, it's noted for attracting hummingbirds, so keep your eyes open for those lovely little creatures.

And, please, watch those caps when typing... it reads as like someone is shouting.

Not sure where in Texas you live, but in Central Texas, the time to fertilize is around mid-April. Yep, you fertilized those weeds.
Best thing to do is start pulling them, the ones with seeds first, followed by the ones with flowers. You can mow, but I strongly suggest you remove any weeds that have formed seeds first.
And don't be tempted by the "Weed and Feed" products. Around here, the time to weed and the time to feed are not the same. And the chemicals from one popular product are now showing up in alarming rates in our ground water.
As Meldy mentioned, Corn gluten will work to prevent seeds from germinating, but you would have had to do that in January or Feb.
Good luck!