Is it a wand core that gives it its awareness?

It is a combination of the core, wood, length & flexibility, and user.

But if anything, the wood might have the most awareness...

Core

Not that hard to parse through the writing on Wand Cores as there are only 3. Each description (as you already noted with unicorn hair) describes the core's "personality". There is a good summary at the beginning taken from Mr. Ollivander's notes:

Readers should bear in mind that each wand is the composite of its wood, its core and the experience and nature of its owner; that tendencies of each may counterbalance or outweigh the other; so this can only be a very general overview of an immensely complex subject.

_{Wand Cores by J.K. Rowling on Pottermore}

Wood

It gets a bit more involved when talking about the woods as there are 38! However, each wood is given a "personality" as well. Again instead of listing them all, here is a summary from Mr. Ollivander's notes.

Also in the introduction it states, "As will be seen, Mr Ollivander believes that wand wood has almost human powers of perception and preferences."

Every single wand is unique and will depend for its character on the particular tree and magical creature from which it derives its materials.

Moreover, each wand, from the moment it finds its ideal owner, will begin to learn from and teach its human partner. Therefore, the following must be seen as general notes on each of the wood types I like to work with best, and ought not to be taken to describe any individual wand.

_{Wand Woods by J.K. Rowling on Pottermore}

Length & Flexibility

This is really the tertiary aspect of the wand, but it still offer a bit of insight into the character of wielder...

However, no single aspect of wand composition should be considered in isolation of all the others, and the type of wood, the core and the flexibility may either counterbalance or enhance the attributes of the wand’s length.

...

Wand flexibility or rigidity denotes the degree of adaptability and willingness to change possessed by the wand-and-owner pair – although, again, this factor ought not to be considered separately from the wand wood, core and length, nor of the owner’s life experience and style of magic, all of which will combine to make the wand in question unique.

_{Wand Length & Flexibility by J.K. Rowling on Pottermore}