  We report the magnetic properties of the double perovskites SrLaCuSbO$_6$
(SLCSO) and SrLaCuNbO$_6$ (SLCNO). The temperature dependence of the magnetic
susceptibilities of both compounds shows a broad maximum characteristic of an
$S\,=\,1/2$ square lattice Heisenberg antiferromagnet. Magnetic ordering occurs
at $T_{\rm N}\,{=}\,13.6$ and 15.7 K for SLCSO and SLCNO, respectively. Neutron
powder diffraction measurements reveal contrasting spin structures in both
compounds. The spin structures of SLCSO and SLCNO below $T_{\rm N}$ are
N\'{e}el antiferromagnetic and collinear antiferromagnetic, respectively. This
result demonstrates that the nearest-neighbor interaction is dominant in SLCSO,
whereas the next-nearest-neighbor interaction is dominant in SLCNO. The
magnitude of the ordered moment was evaluated at 3.5 K to be
$m\,{=}\,0.39(3)\,\mu_{\rm B}$ for SLCSO and $0.37(1)\,\mu_{\rm B}$ for SLCNO,
which are significantly smaller than those calculated using linear spin wave
theory. We infer that the small ordered moment is caused by the effect of
exchange bond randomness arising from the site disorder of Sr and La ions.
