You described their larvae and if you remember the beetles flying around last summer, that is extra confirmation that grubs are likely the problem. However, even if you don't remember them, you probably have white grub problems based on your description. You can take a sample to your local NMSU Cooperative Extension Service office to confirm that you have collected grubs. Another confirmation is the symptom you described, patches of dead grass. While other things can cause this symptom, the presence of the grubs at the edge, where the living and dead grass meet, makes it likely that grubs are responsible.
Grubs eat the roots of the grass, causing the grass to die.
With the roots removed, the grass can often be lifted from the soil like a rug. That is another confirmation. At this time of the year you should do nothing. The grubs are full grown and are not feeding much. They will soon pupate and begin the metamorphosis into June beetles.