Yes, go with the broadleaf weed control. Three-way might be a bit more gentle on the new grass. In my tests it was OK on perennial rye after it attained a height of 3 inches.
Exactly what kind of seed was used? Something suited to you conditions and climate, perhaps. Not too much rye, I hope? Are you in tall fescue country? More suited to Bermuda? Near or after first frost is still OK. There is a high chance of many weed seeds in the soil--normal for weeds to happen after soil tillage and water is applied.
Quicksilver can be applied 7 days after emergence for most grasses; tall fescue may have slight yellowing.
What kind of weedy grass? Annual or perennial? What are your temperatures? Soil temps?
Henbit is a winter annual. If you can't spray it now, it will die in spring after a couple mowings. Oxalis can be either annual or perennial.
Heavy seed will not solve a shade problem. If it was thin due to shade--it will happen again--unless the shade problem is addressed. Reduce the shade...or...plant a shade tolerant ground cover.