Here is a very brief explanation of what is going on. You can find more details in the papers that Ana lists and in arxiv.org/abs/math/0604617 as you say.

If $G$ is a reductive group, the Hitchin base $\text{Hitch}_{G}$ is defined to be the moduli of cameral covers of $X$. Concretely

$$ \tag{1} \text{Hitch}_{G} = H^{0}(X, tot(\mathfrak{t}\otimes \omega_{X})/W)$$

where $\mathfrak{t}$ is a Cartan subalgera in $\mathfrak{g}$. This space is non-canonically isomorphic to

$$ \oplus_{i = 1}^{r} H^{0}(X, \omega_{X}^{\otimes d_{i}})$$

where $d_{1}, \ldots, d_{r}$ are the degrees of the members of a basis of homogeneous invariant polynomials on $\mathfrak{g}$.

It is better to work with the description (1) since it does not involve choices. Given a section $b \in H^{0}(X, tot(\mathfrak{t}\otimes \omega_{X})/W)$ we can construct a cameral cover $\widetilde{X}_{b} \to X$ as the pull back of the natural $W$-cover

$$tot(\mathfrak{t}\otimes \omega_{X}) \to tot(\mathfrak{t}\otimes \omega_{X})/W$$

by the map $b : X \to tot(\mathfrak{t}\otimes \omega_{X})/W$.

The universal cameral cover then is a subvariety

$$ \widetilde{X} \subset tot(\mathfrak{t}\otimes \omega_{X}) \times \text{Hitch}_{G}$$

(rather than a subvariety in $tot(\omega_{X})\times \text{Hitch}_{G}$).

Furthermore the fibers of the Hitchin map are not the relative Picard varieties along the fibers of the universal cameral cover. They are generalized Prym varieties associated with tis relative Picard varieties and the $W$-action. Roughly you will have

$$T^{*}\text{Bun}^{0}_{G} = \text{Hom}(\Lambda,Pic(\widetilde{X}^{0}/\text{Hitch}^{0}_{G})^{W} $$

where $\Lambda$ is the character lattice of a maximal torus of $G$.

However this description is only correct up to isogeny. The precise description is given in the Donagi-Gaitsgory paper and is also recalled in our paper arxiv.org/abs/math/0604617.

Here is a very brief explanation of what is going on. You can find more details in the papers that Ana lists and in arxiv.org/abs/math/0604617 as you say.

If $G$ is a reductive group, the Hitchin base $\text{Hitch}_{G}$ is defined to be the moduli of cameral covers of $X$. Concretely

$$ \tag{1} \text{Hitch}_{G} = H^{0}(X, tot(\mathfrak{t}\otimes \omega_{X})/W)$$

where $\mathfrak{t}$ is a Cartan subalgera in $\mathfrak{g}$. This space is non-canonically isomorphic to

$$ \oplus_{i = 1}^{r} H^{0}(X, \omega_{X}^{\otimes d_{i}})$$

where $d_{1}, \ldots, d_{r}$ are the degrees of the members of a basis of homogeneous invariant polynomials on $\mathfrak{g}$.

It is better to work with the description (1) since it does not involve choices. Given a section $b \in H^{0}(X, tot(\mathfrak{t}\otimes \omega_{X})/W)$ we can construct a cameral cover $\widetilde{X}_{b} \to X$ as the pull back of the natural $W$-cover

$$tot(\mathfrak{t}\otimes \omega_{X}) \to tot(\mathfrak{t}\otimes \omega_{X})/W$$

by the map $b : X \to tot(\mathfrak{t}\otimes \omega_{X})/W$.

The universal cameral cover then is a subvariety

$$ \widetilde{X} \subset tot(\mathfrak{t}\otimes \omega_{X}) \times \text{Hitch}_{G}$$

(rather than a subvariety in $tot(\omega_{X})\times \text{Hitch}_{G}$).

Furthermore the fibers of the Hitchin map are not the relative Picard varieties along the fibers of the universal cameral cover. They are generalized Prym varieties associated with this relative Picard varieties and the $W$-action. Roughly you will have

$$T^{*}\text{Bun}^{0}_{G} = \text{Hom}(\Lambda,Pic(\widetilde{X}^{0}/\text{Hitch}^{0}_{G}))^{W} $$

where $\Lambda$ is the character lattice of a maximal torus of $G$.

However this description is only correct up to isogeny. The precise description is given in the Donagi-Gaitsgory paper and is also recalled in our paper arxiv.org/abs/math/0604617.