You may have a couple of things going on at the same time.  First, they are probably stressed due to the hot weather.  Second, it's good that they rebounded after watering, but daily watering shouldn't be necessary.  Plants will show the same stress symptoms, at first, to over and under watering, which is frustrating, especially as a new gardener. 

It looks like the lilacs are planted in the ground, but you said the roots are dry.  Are you digging a hole in the soil to feel the roots or are you assuming the roots are dry because the leaves are wilting?  If the surrounding soil is wet, the roots are wet.  Give the soil a chance to dry out a little, it should feel as moist as a squeezed out sponge in between waterings.  Before you water again, dig a hole about 3 or 4 inches deep that is a couple of inches away from the plant and grab a handful of soil.  Squeeze the soil. When you open your hand it should just start to fall apart.  If you have clay soil, it may not fall apart but it shouldn't feel really wet.  Only water if it feels almost dry. 

Before you cut off the dying leaves, cut a couple of inches off the top of the dying branch.  If it's still green in the stem, it can recover.  If it's brown, or snaps off easily, then cut a few inches at a time, in case there's still some life left in it closer to the main stem.  

In addition, move the mulch about an inch away from the base of the plant.  You do want to use mulch to keep the moisture in the soil, but it shouldn't be right next to the plant.

I hope this helps.  Gardening is a lot of trail and error and some experimentation.  Your lilacs should recover, but it may take a little bit of time.  

I am checking the soil and roots with a moisture tester. The soil about a foot from the plants say 9-10, a few inches from the branches say 1-2. Thank you for your time and response. I will put your knowledge to good use.