My new Gabor trumpet vine…
...had no water for a week in the hot, late June South Carolina sun. It didn't rain all week. The base of the vine still has green leaves, but the vines have a greenish-brown hue and dead leaves. Should I cut down the vines? Should I wait and see if the vines come back after a good watering? I just planted this vine in March, so it's a new plant.
I would suggest water, wait and see.
But you can assess dead vs living stems by bending them; if they are completely dried up and snap off readily they can be considered dead. And/or if you slice some bark off and look for a green growing layer under the bark, it indicates that the stem may still be viable and will leaf out again.