  Particle-decaying states of the light nuclei $^{11,12}$N and $^{12}$O were
studied using the invariant-mass method. The decay energies and intrinsic
widths of a number of states were measured, and the momentum correlations of
three-body decaying states were considered. A second 2$p$-decaying 2$^+$ state
of $^{12}$O was observed for the first time, and a higher energy $^{12}$O state
was observed in the 4$p$+2$\alpha$ decay channel. This 4$p$+2$\alpha$ channel
also contains contributions from fission-like decay paths, including
$^6$Be$_{g.s.}$+$^{6}$Be$_{g.s.}$. Analogs to these states in $^{12}$O were
found in $^{12}$N in the 2$p$+$^{10}$B and 2$p$+$\alpha$+$^6$Li channels. The
momentum correlations for the prompt 2$p$ decay of $^{12}$O$_{g.s.}$ were found
to be nearly identical to those of $^{16}$Ne$_{g.s.}$, and the correlations for
the new 2$^+$ state were found to be consistent with sequential decay through
excited states in $^{11}$N. The momentum correlations for the 2$^+_1$ state in
$^{12}$O provide a new value for the $^{11}$N ground-state energy. The states
in $^{12}$N/$^{12}$O that belong to the $A$=12 isobaric sextet do not deviate
from the quadratic isobaric multiplet mass equation (IMME) form.
