Finding something decent to grow under a Live Oak is one of those eternal challenges for the Austin area. As such, these three question/answer pairs from the Mr Smarty Archives contain a pretty good selection of advice towards native plants that will cope with the shade, alleleopathy and dense roots under a live Oak.
For the other half of your yard, you didn’t voice an explicit question – BUT – Mr Smarty Plants noticed that mention of Jasmine grass. We suspect you mean Asiatic Jasmine [Trachelospermum asiaticum] which is an import from Asia, specifically Korea and Japan. As Wildflower Center aficionados, we feel honor bound to advise you to get rid of that non-native and replace it with a good native. There is a good selection of native groundcovers and vines that have similar growth habits. Consider the perennial herbs: Phyla nodiflora (Texas frogfruit) or Calyptocarpus vialis (Straggler daisy), and the vines: Parthenocissus quinquefolia (Virginia creeper) or Ipomoea cordatotriloba var. cordatotriloba (Tievine).
Finally - "Should you Xeriscape" [note spelling!]. Yes, most certainly, if that is a type of look for your yard that you would appreciate. It clearly is a strong approach if you like the low-maintenance aspect and there is an excellent selection of native plants that are well adapted to the kind of rainfall and sun that we see around here. One of the better solutions for the area under an Oak is a type of xeriscape. This is to use a cover(plastic, cardboard) to suppress the suckers, then cover that with a thick layer of mulch on top. Here are a couple good links I read when I searched "Xeriscape" - Their questions are remarkably like yours!