As other posters have stated, it really comes down to how you define a soul in your world. From an athiestic point of view, people have been soulless forever (at least in any metaphysical sense) and have been doing just fine as human beings. That perspective changes with a religious worldview, and likely depends on how that religion views the soul.
If you're going to make souls a canonical part of your world, you'll need to establish what they do and what they ARE for a person. In my own fantasy setting, souls aren't the essence of a person so much as an imprint that they leave behind (switching around the typical cause and effect) while still having important physical and mental implications, in particular the ability to wield magic. Disrupting or damaging someone's soul in my setting has harsh biological ramifications and inhibits magic use... at least until it reforms.
If you set up souls such that they're the driving force of the human mind (classic dualism) then you've basically just turned your MC into a vegetable. Not so interesting.
You could go with the idea that the soul represents Ego versus the material Id of the physical brain... in which case, not having a soul would make one very instinctual, very emotional, and not particularily inclined to forethought, inhibition or rationality. In this case, they'd be perhaps even MORE inclined to love, enjoy sensual experiences like music, etc.
However, from what you've got so far it seems like you might even want to flip that around, and have the soul represent a person's ID instead. A soulless person would then be extremely detached from all the parts of their mind that they share with the animal kingdom... they would be cold, rational, and indifferent to base pleasures and emotions. Robots, basically.
Yet another possible approach would be to discard the idea that it has to affect those fundamentals, and turn the soul into the part of the mind which connects with supernatural forces in your world. Especially if supernatural influence is ever-present or important somehow, this could manifest as a feeling of loss or emptiness, possibly making the character depressed or apathetic.
Ultimately, it rests on how you define a "soul" and what it does. Nobody in the real world has ever really described that in detail, so you've got a great deal of freedom in making it your own. In fact, you can make it correspond to whatever topic you wish to explore... it's a great way to ask the question "what would happen if the human mind lacked x, y, and z?"