  We re-investigate the magnetically frustrated, {\it
diamond-lattice-antiferromagnet} spinels FeAl$_2$O$_4$ and MnAl$_2$O$_4$ using
magnetization measurements and diffuse scattering of polarized neutrons. In
FeAl$_2$O$_4$, macroscopic measurements evidence a "cusp" in zero field-cooled
susceptibility around 13~K. Dynamic magnetic susceptibility and {\it memory
effect} experiments provide results that do not conform with a canonical
spin-glass scenario in this material. Through polarized neutron scattering
studies, absence of long-range magnetic order down to 4~K is confirmed in
FeAl$_2$O$_4$. By modeling the powder averaged differential magnetic neutron
scattering cross-section, we estimate that the spin-spin correlations in this
compound extend up to the third nearest-neighbour shell. The estimated value of
the Land\'{e} $g$ factor points towards orbital contributions from Fe$^{2+}$.
This is also supported by a Curie-Weiss analysis of the magnetic
susceptibility. MnAl$_2$O$_4$, on the contrary, undergoes a magnetic phase
transition into a long-range ordered state below $\approx$ 40~K, which is
confirmed by macroscopic measurements and polarized neutron diffraction.
However, the polarized neutron studies reveal the existence of prominent
spin-fluctuations co-existing with long-range antiferromagnetic order. The
magnetic diffuse intensity suggests a similar short range order as in
FeAl$_2$O$_4$. Results of the present work supports the importance of spin-spin
correlations in understanding magnetic response of frustrated magnets like
$A$-site spinels which have predominant short-range spin correlations
reminiscent of the "spin liquid" state.
