A healthy vigorous stand of turf grass will keep most of the poa anna out. First make sure that the weed is poa anna, and not a perennial cool season grass. Poa anna will not usually make it through the summer in your area. However, you must fix the underlying problems to keep the poa anna out of your yard. Does the bermuda get enough nitrogen fertilization at least a couple of pounds of nitrogen per 1000 square feet during the growing season. You said that the problem side of your yard has a lot of shade are tree roots also competing for water in this area? Also, bermuda grass needs at least 6 to 8 hours of direct sun a day. If that can not be achieved then maybe you need a more shade tolerant turf, such as zoysia, to be able to compete with the poa anna. If the area stays wet it might also be highly compacted so aerifiying during the summer would be a good idea. If some or all of these problems can't be corrected then you should just live with the poa anna because once you get rid of the poa anna another weed will take it's place if you don't have a healthy turf grass to fill in the area that the poa anna occupied. Eradicating the winter annuals from bermuda (poa anna, chickweed, and henbit) and keeping it out is not a difficult task.You can spray Roundup, buy the concentrate that contains just glyphosate and spray at the low rate, on your dormant bermuda grass at the end of January beginning of February.This is not as scary as it sounds just make sure you don't have any green tissue in the leaves, stems, and stolons. Maybe start small and just spray a couple of spots this year and see how it turns out. As for the henbit and chickweed, Bayer has a broadleaf product with 2,4-D, dimethylamine salt/ Mecoprop-p, dimethylamine salt/ Dicamba, dimethylamine salt. Spray this product mid/late spring when air temps are over 50 degrees. Pendimethalin is the best pre emergent option for the home owner, to keep out poa anna. Pendimethalin is sold as Halts by Scotts. Follow the labeled rate and apply around August 15th water in immediately with half an inch of water. Pendimethalin will also take care of your henbit and chickweed as well.

Can you tell us what region you live in and what weeds you're finding in your yard?

I'm in Greenville SC, so not too different of a climate zone. Fall of 2014, I had a similar situation as yours, but much worse. Sodded Bermuda with a massive number of broadleaf and grassy weeds (maybe about 3x what your pics show). Battled those down all of 2015 and did a lot of reading and research to figure out what my weeds were and what was the best way to make the next year better. I went with an application of Prodiamine, which is the same active ingredient in Stonewall. I had planned to do a split application, but decided the 2nd app wasn't needed because of our mild temperatures. So, it was about 0.2oz/1000 sqft applied Sept 12th 2015 with my sprayer. I went with the liquid for better coverage. This was my first time using it, and I am very impressed with the results. I have almost no broadleaf weeds so far this year *except* right around the edges of the lawn where I probably didn't spray well. In your 2nd pic, I believe the weed on the right is a Carolina Geranium, and the middle one is some sort of thistle (Creeping Thistle?). Not sure about that left one, however, Prodiamine should have fairly consistently knocked out those first two. That makes me think either the application wasn't enough, wasn't timed correctly, or wasn't spread evenly. So, if I had to suggest a change for next year, you might try the liquid instead of the granules, and apply about a month earlier. Target Sept 15th for our area, but with Prodiamine, it lasts a good amount of time in the soil, so earlier won't be a bad thing.

If you are overseeding with bermuda seed, that window closed about a month ago. Bermuda seed needs hot, Hot, HOT soil to germinate. The real question is why do you have thin bermuda? If the area is in the shade, no amount of bermuda seed will help. The only thing that will help is to get rid of the shade. If you can't get rid of the shade, then consider using a different ground cover for the thin areas. St Augustine grows well in the shade but will invade the bermuda in the sun. There are some zoysia varieties that look more like bermuda and will thrive in the shade. Or you could take out the bermuda in those thin spots and make flower gardens. If you are overseeding with rye to make it green for the winter, please don't. Rye in the spring will shade out the bermuda as much as trees will. Often the rye overseed leads to thin turf for the following spring and summer.