Investigating the cause of craving (multiple questions)

Recently I asked myself, "If craving causes dukkha, then what causes craving?" I remembered the chain of dependent origination, and I noticed a few things:

  1. Ignorance is the root cause of the chain. IIRC this is ignorance of the four noble truths (and not simply a lack of intellectual understanding). However, the third chain is consciousness, which confuses me because, if eliminating ignorance eliminates the chain, then wouldn't removing ignorance mean a removal of consciousness? Buddha was conscious after his enlightenment, so I feel I am misunderstanding something here.

  2. Before craving (tanha) in the chain is sensation (vedana). As long as we are conscious, there are going to be sensations, so how does one prevent craving from following sensations?

  3. After craving in the chain is clinging (upadana). Now I can see how this point of the chain directly involves will; if we don't cling to craving, then we break the cycle. However, it's not clear from the chain as-it-is that refusing to cling will end craving once and for all; it seems only ending ignorance (or sensations) would do that, which leads me back to my first couple of questions.

How do I make sense of all of this?