You are wise to concern yourself over the possible long-term problems of shade trees too closely spaced. There is a very good tree guide published by the City of Austin. It provides information on the mature size of various common trees and will be a great help in choosing replacement trees. Further information on tree species you consider can be found by entering each tree's name in the Lady Bird Johnson Wildflower Center Plant Database. This latter source sometimes gives a slightly different estimate of mature sizes.
One possibility would be replacing you ash with another ash, namely Fraxinus albicans (Texas ash). Or, if you wish trees that will pretty much fill the canopy space, consider Quercus buckleyi (Texas red oak), Ulmus crassifolia (Cedar elm) or Acer grandidentatum (Bigtooth maple). The maple is becoming more popular, especially if your setting is somewhat moist. But rumor has it that the maple does not like city mains water, so placement where rainwater can occasionally soak the ground is desirable. A smaller oak like the Quercus laceyi (Lacey oak) would give you more open space. Two even smaller trees, Cercis canadensis var. texensis (Texas redbud) or Chilopsis linearis (Desert willow) cast a thinner shade if that is desirable.
Many of the tree species shown on the City of Austin guide are available from local plant nurseries. The best time for planting is winter, when leaves have dropped and water requirement is minimal. Our web site has a good guide for tree planting.