So you asked about a good cover crop that will fix nitrogen, add carbon
to soil and choke out weeds. The best option is to come up with a variety of plants to do these jobs. We have already covered the clovers. White clover is a good soil cover and nitrogen fixer, does fairly well in low moisture areas and can be terminated fairly quickly.
Red clover is similar but produces more growth and is harder to terminate. I am not certain how it stands up to drought in your area, but by me it is impervious. Alfalfa is really tops for producing biomass and produces a deep taproot but might be hard to terminate.
As for grasses, there is a huge variety of options, but I would consider an annual as it self-terminates. One grass to consider is wheat. If wheat works for you, you will have to decide if spring or winter wheat is appropriate, or maybe a mixture of both. I suggest wheat because seeds thrown on the ground will sprout on their own, grow, produce their own seeds, die and re-seed themselves. Wheat can be mowed for control. Also, the list of grasses is long and varied and others can surely offer good or better options. Worth considering is planting a C4 plant as it will be a far better converter of sunlight to carbon than a C3 plant (clover and wheat are both C4 plants). Corn and sorghum are both C4 plants. There are others.
At any rate, if you glean information on this site, I am sure that you can find a decent seed mixture.