Verlinde formula, moduli space vector bundle on genus 2,3 curves.

Question

I'd like to prove "by hands" the Verlinde formula for moduli space of rank two semistable vector bundles with fixed determinant on a curve of genus two and three. For a curve of genus two and even degree I proved that this moduli space is isomorphic to $\mathbb{P}^3$, using GIT techniques, so it's ok! For the odd case I know that this space is isomorphic to the intersection of two quadric in $\mathbb{P}^5$. For what concerne the proof I saw the Newstead's article. Does a modern proof of this statement exist? Furthermore, could you give me a suggestion about how to proceed the genus three case? I studied the singularity and went on with three blow up for the even case following the work of Kiem. How can I say in this way something about the cohomology of this space?