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We present a study of six dusty and gaseous pillars (containing the HH 1004 and HH 1010 objects) and globules (that contain the HH 666, HH 900, HH 1006, and HH 1066 objects) localized in the Carina nebula using sensitive and high angular resolution ($\sim$0.3$''$) Atacama Large Millimeter/Sub-millimeter Array (ALMA) observations. This is a more extensive study that the one presented in \citet{Cortes}. As in this former study, we also analyzed the 1.3 mm continuum emission and C$^{18}$O(2$-$1), N$_2$D$^+$(3$-$2) and $^{12}$CO(2$-$1) spectral lines. These new observations revealed the molecular outflows emanating from the pillars, the dusty envelopes$+$disks that are exciting them, and the extended HH objects far from their respective pillars. We reveal that the masses of the disks$+$envelopes are in a range of 0.02 to 0.38 M$_\odot$, and those for the molecular outflows are of the order of 10$^{-3}$ M$_\odot$, which suggests that their exciting sources might be low- or intermediate-mass protostars as already revealed in recent studies at infrared and submillimeter bands. In the regions associated with the objects HH 900 and HH 1004, we report multiple millimeter continuum sources, from where several molecular outflows emanate.