The weeds you have should pluck out relatively easily in moist soil. Some of those have embarrassingly weak root systems. If you have huge areas of the weeds and want to spray them, some of them react quickly to vinegar spray.

I have seen Dutch white clover growing in the Texas Panhandle but it wasn't doing very well. Of course even the bermuda was having trouble in that turf(!). I have read here that clover likes an acidic soil, which you probably don't have in DFW. I would like to read about Texas clover installations to see if anyone is having good luck with it.

I do know of a variety of zoysia, called Shadow Turf, that will grow very well in the shade. It is aggressive and will invade flower or veggie beds unless you have a solid barrier at least 3 inches deep and an inch or so above ground. It is expensive - one set of plugs (64 plugs?) will cost you about $90, but like I said, it is aggressive and will spread out until it is physically stopped. If you can find some at a nursery that looks pretty good, now is a great time to install it. I planted Shadow Turf last May and will be watching it carefully this year as it competes with bermuda to take over a patch in my stone driveway. I have a couple pictures of it growing in deep shade. I need to figure out a link for those so you can see.