Question
I scratched the bark like you suggested to check if the tree is still alive.  It appears to be brown under the bark.  Since it still has a few leaves that look alive, I snipped the end of a branch.  It is not brown all the way through, but has what appears to be bark around a brown inner layer and then the middle was a whitish green color.  Is that what I should see or is the tree a lost cause?

The avocado is definitely green and looks like it may already have new growth.  How long should I wait before applying any fertilizer?

Answer
Give the Lychee some time, it may be on the cusp, but the Avocado should be fine, and won't require anything but irrigation for the next couple of months; around early April apply a well balanced Citrus fertilizer (about a cup) spread evenly from the drip line in. If the Lychee doesn't show any new growth in the next 2 weeks, then start bending some of the branches, if they just snap off and there is no sign of just bending instead, then it is indeed gone. Nick