  The $^{12}$C+$^{12}$C reaction at 95 MeV has been studied through the
complete charge identification of its products by means of the GARFIELD+RCo
experimental set-up at INFN Laboratori Nazionali di Legnaro (LNL). In this
paper, the first of a series of two, a comparison to a dedicated
Hauser-Feshbach calculation allows to select a set of dissipative events which
corresponds, to a large extent, to the statistical evaporation of highly
excited $^{24}$Mg. Information on the isotopic distribution of the evaporation
residues in coincidence with their complete evaporation chain is also
extracted. The set of data puts strong constraints on the behaviour of the
level density of light nuclei above the threshold for particle emission. In
particular, a fast increase of the level density parameter with excitation
energy is supported by the data. Residual deviations from a statistical
behaviour are seen in two specific channels, and tentatively associated with a
contamination from direct reactions and/or $\alpha$-clustering effects. These
channels are studied in further details in the second paper of the series.
