Weed-B-Gone is one, but it's also in (for example) Ace-branded weed products. Check the label.
There's a contingent here who really dump on the weed-n-feed approach. I don't think it's out of place for someone who is trying to get a problem under control. Your lawn probably needs both fertilizer AND weed control if you have a problem this bad, and some of both is better than nothing.
As you get more in tune with your lawn you can refine your methods.
In any case, ONE WEEK is far too soon to expect results. The "weed" part you put down will suppress new growth and small plants but will have little effect on the maturing dandelions ready to sprout their glorious yellow flowers (the pests). You'll still need to go around pulling those (by the roots, but you know that), or at least using a spot sprayer. It will also prevent the seeds from germinating in high numbers, over time. If you followed the instructions on the package, likely you used about half of what you need and will apply again in 30 days' time, which will be a sucker punch for whatever got through the first application.
Don't expect miracle results this year. You're merely getting the problem under control. Next year, you'll see many fewer dandelions and healthier grass, and you may be able to deal with it with a spot spray or pulling approach. If it's still pretty daunting go with another round of weed'n'feed. It may take two to five years of diligent maintenance to get rid of every last dandelion, though! Think long-term.
Almost any lawn can be improved. Weeds will be a smaller problem in healthy lawns. It's often said that the biggest problem is people just mowing too short. Two inches should probably be a MINIMUM, because lower cutting just damages the grass and cuts off too much of the fresh growth. (Different varieties of grass have different comfort heights, too.) Taller grass lets less light get to the weeds, too, and the cuttings (if you use a mulcher mower) create a good nitrogen-rich natural fertilizer at the root level.
Reseeding is best done in the fall, of course, but there's no reason you can't using a patch approach in spring or early summer. You just have to really baby those patches -- water twice daily at first, for instance, and reduce that when you see sprouts to daily, then to every couple of days (depending on precipitation). Make sure you till in some starter fertilizer in the larger bare spots before seeding.
Bare spots from not watering probably mean you need a hardier, sunny-type grass in those areas. A healthy lawn can do just fine most all summer with minimal watering -- it's called going dormant, and is the natural summer state of grass. When you vigorously water it early in the season, though, it comes to expect that all summer and doesn't put down a deep enough root system. Are you *deep* watering or just sprinkling?