Where do you live? I'm curious as to why your bermuda is still dormant.

The stuff with the white flowers is poa annua. When bermuda is truly dormant you can spray the poa with Roundup. If the bermuda is starting to return, the Roundup will wipe it out.

The other weed should come out easily with the True Green herbicide.

Just because you have TG doing fertilizer and weed control, the most important part of the lawn (by far) is still your responsibility. Those would be watering and mowing. Even if you have a lawn mower every week you still have to water. Here's how.

Watering: Deep and infrequent is the mantra for watering. This is for all turf grass all over the place. Deep means 1 inch all at one time. Put some cat food or tuna cans around the yard, and time how long it takes your sprinkler(s) to fill all the cans. Memorize that time. That will be the time you water from now on. My hose, sprinkler and water pressure takes 8 full hours to fill the cans. Your time will likely be less. I like gentle watering. As for watering frequency, that depends on the daytime air temperature. With temps in the 90s, deep water once per week. With temps in the 80s, deep water once every 2 weeks. With temps in the 70s, deep water once every 3 weeks. With temps below 70, deep water once a month. Note that you have to keep up with quickly changing temps in the spring and fall. This deep and infrequent schedule works in Phoenix and in Vermont, so it should work for you. The reason for deep and infrequent is to grow deeper, more drought resistant roots and to allow the soil to dry completely at the surface for several days before watering again. If it rains, reset your calendar to account for the rainfall.