Cedar ElmUlmus crassifolia (Cedar elm) with its glossy green leaves, smaller than those of American Elm, is an attractive landscape plant. (more info)
Mr. Smarty Plant has found that is is very hard to diagnose tree problems from a written description, but you have given some clues that will allow some reasonable guesses. Along with my guesses, I will provide some links that will help you determine what the problem may be.
The symptoms you describe; yellowing , dry leaves with the top-most leaves showing evidence of a chewing insect perhaps, could be caused by several things, but Dutch Elm Disease is probably not one of them.
Dutch Elm Disease (DED) is caused by the fungus Ophiostoma novo-ulmi which is introduced to the tree by the native elm bark beetle, Hylurgopinus rufipes and the lesser European bark beetle, Scolytus multistriatus.
The first symptom observed in American elm is yellow foliage on one or more branches, from late spring to midsummer. The affected foliage soon becomes wilted and brown; this symptom is called "flagging." This doesn’t sound like what you described. These three links have more information about the disease.
University of Illinois
Yellowing can be a symptom caused by an aphid infestation, which seems to be common in Austin this summer. This link from UC Davis describes the pests and possible control measures.
Another pest that could account for the yellowing is spidermites. Reading the two links below could make you a neighborhood expert on these little plant suckers.
Colorado State University
UC Davis
Bacterial leaf scorch can cause yellowing, but the images shown in these two sources don’t look to me like what you described.
Bartlett Tree Research
The United States Nattional Arboretum
Since aphids and spidermites lack chewing mouth parts, we can’t blame the chewed leaves on them, however a possible culprit could be the Elm leaf beetle. The two links below give details about these destructive bugs.
Colorado State University
University of Illinois
The bottom line is that you need someone knowledgeable about tree sto take a look at your elm tree to make an assessment of the situation. This person could be a Certified Arborist , or someone from the Travis County Office of Texas AgriLife Extension.