  Travelling solitary waves in the one-dimensional discrete nonlinear
Schr\"{o}dinger equation (DNLSE) with saturable onsite nonlinearity are
studied. A variational approximation (VA) for the solitary waves is derived in
an analytical form. The stability is also studied by means of the VA,
demonstrating that the solitons are stable, which is consistent with previously
published results. Then, the VA is applied to predict parameters of travelling
solitons with non-oscillatory tails (\textit{embedded solitons}, ESs).
Two-soliton bound states are considered too. The separation distance between
the solitons forming the bound state is derived by means of the VA. A numerical
scheme based on the discretization of the equation in the moving coordinate
frame is derived and implemented using the Newton--Raphson method. In general,
a good agreement between the analytical and numerical results is obtained. In
particular, we demonstrate the relevance of the analytical prediction of
characteristics of the embedded solitons.
