You can't know how much trouble you saved yourself by writing in here before doing anything else.

Your worst problem is you have dallisgrass, not crabgrass. Crabgrass would die normally at the first frost, but not dallisgrass. It is a perennial and will take over quickly if you don't stop it. Since it is a grass, anything you use to kill it will also kill your bermuda. But since you have bermuda, it will grow back without doing anything else.

Your second worst problem is the St Augustine weed. It, too, will take over under certain conditions. You don't have to do those, but unless you go the extra mile, a properly cared for bermuda lawn infested with St Aug will always look weedy. Possibly the easiest way to get rid of it is to make it sick. That's easy because all you have to do is cover it with some newspaper and wet it down every night. Leave the paper on for about a week and you should start to see lesions on the grass blades that look like the spots on this grass.

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Once you see that you can peel the paper off and let Nature take its course. The disease will spread throughout the St Aug and wipe it out. Since it does not come back from seed, once it's gone, it's gone. Bermuda might get the disease, but St Aug gets it fast and furious. It will die out in a month or so. If you get a freeze before it all dies out this year, the disease will come back next spring to get the rest of it. The bermuda will fill in as fast as the St Augustine dies out.

Now the problem you don't know you have, and the problem that could easily become your biggest problem, is seeding into your existing bermuda. DO NOT DO THAT. Bermuda needs no seed. Once you have it, you practically have it for life. The problem with seeding into it is that the seed is very nearly a different species of plant from the grass you already have. What you have is a very good hybrid called Tif 419. What you get in seed is a variety of common bermuda seeds. The two plants do not mix well together. And I happen to have a picture of that, too.

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The fine bladed grass is Tif 419. The coarse bladed stuff running over top of the Tif is an invasion of common bermuda. As with St Augustine, once you have a mix of Tif with common, it always looks weedy.

Find the Bermuda Bible online and memorize it. There are things you should be doing all the time. I can summarize:
Every month: fertilize with a high nitrogen fertilizer
Every week: water deeply
2x per week: mow
But get the BB, it will explain when not to do those things.

For your shady area you do want St Augustine. If you get a dwarf variety you can mow at the same height as the bermuda. If you cannot get a dwarf, then it is best to mow St Aug at the mower's highest setting (4 inches). St Aug is much less intensive to care for:
Memorial Day: fertilize with a regular fertilizer (not weed-n-feed)
Every week after the temps get high: water once, deeply, and mulch mow high
Labor Day: fertilize with a regular fertilizer
Thanksgiving: fertilize with a regular fertilizer

By the way, centipede grass is out of the question in Dallas' alkaline soils.