Tzava'ath HaRivash : Sexual desire is the source of all ta'avotHere's my attempt to understand how they interact and complement one another rather than contradict: (note that above I've already paraphrased them in a way of least possible conflict)
Noam Elimelech : Ga'avah is the force behind all aveirot
Noam Elimelech : Eating is the universal tikkun
Sexual desire is the origin of all desire. Freud says something similar, but that doesn't prove our point. At its essence the sexual desire is to unite two parts of a single whole. All of our desires begin with a perceived lack. We see something that promises to fill that emptiness within us and that is why we desire it. The catch with desires is that we feel them, but we don't generally understand why we feel them.
Ga'avah is the basis of all sin, because we need to see ourselves as separate and important before we would sin. If we were to be aware of our life as originating from God and His mercies, we wouldn't have the audacity to sin. If we recognized the harm our actions do to ourselves and others, we also wouldn't sin. It is only when we refuse to acknowledge our source in HaShem and claim a greater importance over others that we could sin.
Eating is the basis of all tikkun. How is this so? Eating educates us in a way that solves both of our major failings at once. On the one hand, eating is one of our first challenges as living beings that allows us to become consciously aware of how our lacks, our inner emptiness, affects what we desire. On the other hand, eating forces us to recognize that we are not complete and independent. This recognition opens a door for us to walk through humbly, acknowledging all of HaShem's mercies and recognizing the needs of others as well.
When we reach complete consciousness in our eating habbits, we reach a level of awareness from which we can also vanquish our desires and acknowledge HaShem's constant kindness in filling our needs.