  In this paper we consider certain rigidly rotating closed string
configurations in an asymptotically non-AdS string background. The string
background is a deformation of $AdS_3 \times {\cal M}_7$. It interpolates
between $AdS_3 $ and asymptotically linear dilaton ${\rm I\!R} \times S^1
\times {\rm I\!R}$ spacetime (times the internal compact manifold ${\cal
M}_7$). We compute the quantity $E - J$ (in the large $J$ limit) where $E$ is
the energy and $J$ is the angular momentum of the spinning strings. In the two
dimensional CFT dual to string theory on $AdS_3$ (times ${\cal M}_7$) it gives
the anomalous dimensions of certain twist two and higher operators. We show in
the deformed background that $E - J$ is bounded. At a special value of the
deformation coupling we also show that for spinning closed strings containing
$n > 2$ cusps or spikes both $E$ and $J$ are bounded. In the CFT dual to string
theory on $AdS_3$ (times ${\cal M}_7$) the spinning cusped strings describe
operators with twist $n$ larger than two. In general, at other values of the
deformation coupling, we demonstrate that this feature is exhibited only by
those cusped strings with $n > n_0$ where $n_0$ is determined only by the
deformation coupling. We also give simple exact Regge relations between $E$ and
$J$. We also study the closely related cusp anomalous dimension of a light-like
Wilson loop. We comment on what $E - J$ measures away from the CFT along the
deformation in the coupling space. In the long string sector the deformation is
dual to a single trace $T{\bar T}$ deformed orbifold theory. We determine the
associated deformed sinh-Gordon model that classically describes the (long)
strings near the boundary. This provides an example of single trace $T{\bar T}$
deformation in non-orbifold theories.
