Cliquewidth of Almost Cographs
I will try to answer this old question of yours, although I am not sure my answer is conclusive but it should point you in the right direction.

First let us discuss linear clique-width.
If a graph has linear clique-width $k$, and one add $1$ vertex to the graph, that vertex can always be placed first in the ordering with a unique color.
Hence linear clique-width only increases by at most 1 when you add a vertex.

Gurski and Wanke showed in "On the relationship between NLC-width and linear NLC-width" that cographs have unbounded linear clique-width.

Since cographs have unbounded linear clique-width but bounded clique-width any good clique decomposition must have a tree structure.
We must show that we can force arbitrarily many deep branches. 
Now we do as we do for trees, construct a tree with at 2^k leaves add k vertices and each leaf is connected to a unique subset of new vertices.