Rearranging strings so that the Hamming distance between them is 1
Grid graphs are finite node-induced graphs of the infinite planar grid (whose vertices are integral points, and two vertices are linked when their Euclidean distance is 1). A. Itai, C. H. Papadimitriou, and J. L. Szwarcfiter proved that the Hamiltonian path problem on grid graphs is NP-complete.

To reduce that to your problem, suppose the grid graph has $m$ rows and $n$ columns. The $i$-th row is labeled with the string $a^ib^{m-i}$, and the $j$-th column is labeled with the string $a^jb^{n-j}$. Then a vertex corresponds to the concatenation of the two labels of its row and column. Now it is easy to show the unit disk graph under Hamming distance over all these concatenated strings is exactly the same as the original grid graph.