Weeds way out of control

I have a back yard that just went to the weeds. I used all the right products and did all the right things. I know I did it right because my front yard looks fine except under a tree where the grass is slow to grow but is filling in.
However, in the back yard, I have stickers of all kinds, johnson's weed, and other weeds. The main weed which destroyed the lawn is a weed which grows extremely quickly. I have seen this weed get up to 6 to 7 foot tall unchecked. The stickers are in strings which grow out of the top. This weed went from a few in the yard to taking over and chocking out the grass I was growing. Its main stalk can be anywhere from 1/2 inch to 2 inches thick and be 3 foot high in a matter of 2 weeks.
In the spring, I had graded the back yard and seeded it. Grass began to grow. Then within 2 weeks I started to get this weed. the more I killed it the faster it seemed to grow. Where I had one suddenly I had a patch. The weed has taken 95% of the lawn.
The weed killers I want to use will kill the little grass I have and may prevent grass growth.
I have had a lawn care person tell me I should regrade and till the yard and put down a new type of grass which grows extra fast. I saw it on TV once but have not seen it again and can not remember the name. This grass is supposed to grow anywhere even on cinder blocks which was part of the TV ad. I was told that the weed I have is spread by birds and if I have open dirt they will start up. That I needed to either roll out grass which is extremely costly or buy this extremely fast growing grass. Supposed to come in plastic jar of seeds which covers 100sq ft.
Any suggestions? I do not mind using this type of grass but I would rather know how to stop this weed problem now and not have to deal with it all winter long. I live in West Texas where the grass can grow all winter. There is little dormant time. The grass may remain dormant for as little as a 2 to 3 weeks until late December/ January. Feburary is the only month the grass is totally dormant. We just do not see much of a winter here. maybe a few days here or no more than 2 weeks there with almost no snow fall to speak of.
I just want to get rid of the weeds in a cost effective way. I say this as I have several properties and I have never had this problem before. With other weed problems I have been able to either spray the weeds gone or hire two men for $40 to pull all the weeds and straiten up the flower beds. Grass of choice is plain bermuda. second choice is carpet bermuda which is way softer to the feet but has a lighter green to lime color.
However, this problem is resisting the roundup, weed be gone and others I have tried. It kills them but new shoots replace the ones I killed. Within days its like I did nothing at all. So HELP!
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