There are no native turf grasses that grow well in mostly shade. If you area gets 6 or more hours of sun each day, then the Native Lawns: Multi-species of Bouteloua dactyloides (Buffalograss), Bouteloua gracilis (Blue grama) and Hilaria belangeri (Curly mesquite grass) or buffalograss alone (Native Lawns: Buffalograss) would be your choice. With much less than 6 hours of sun per day, however, these grasses aren't going to grow very well at all. Sedges are probably your best bet for the area if you are looking for a lawn-like look. Here is an article, Sedge Lawns for Every Landscape, by John Greenlee. The native sedges for your area that do well in the shade and are evergreen and drought tolerant are:
If you can tolerate taller grasses, there are several native ones that are attractive and do well in shade or partial shade:
There are a couple of low groundcovers that are semi-evergreen and will tolerate light to moderate foot traffic that will also grow in partial shade or shade:
You are absolutely correct that this is a very bad time (because of the drought and extreme heat) to try to get grasses, sedges or groundcovers started either by seed, by sod, or by plants. Whatever you plant is going to require lots of water to establish it and maintain it unless we start getting some significant rainfall soon.
You can look for nurseries that specialize in native plants in your area by searching in our National Suppliers Directory.