One usually defines $w_\alpha(\varepsilon) = x_\alpha(\varepsilon) \cdot x_{-\alpha}(-\varepsilon^{-1}) \cdot x_\alpha(\varepsilon)$, which is the image of
$$\begin{pmatrix} 0 & \varepsilon \\ -\varepsilon^{-1} & 0 \end{pmatrix}$$
under the mapping of $\operatorname{SL}_2\to G$ defined by $x_\alpha$ and $x_{-\alpha}$. Then there are relations (sometimes called Steinberg relations) on $x_\alpha$, $w_\alpha$ and $h_\alpha(\varepsilon)=w_\alpha(\varepsilon)w_\alpha(1)^{-1}$, among them
$$w_\alpha(\varepsilon) \cdot x_\beta(\xi) \cdot w_\alpha(\varepsilon)^{-1} = x_{w_\alpha\beta}\left(\eta_{\alpha\beta} \cdot \varepsilon^{-\langle\beta,\alpha\rangle} \cdot \xi\right).$$
Here the numbers $\eta_{\alpha\beta}=\pm1$ satisfy the following properties:
$$\eta_{\alpha\beta}=\eta_{\alpha,-\beta}, \qquad \eta_{\alpha\alpha} = -1, \qquad \eta_{\alpha\beta}\eta_{\alpha,w_\alpha\beta}=(-1)^{A_{\alpha\beta}},$$
where $A_{\alpha\beta} = 2(\alpha,\beta)/(\alpha,\alpha)$ are the Cartan integers. Moreover,

- $\eta_{\alpha\beta}=1$ if $\alpha\pm\beta\neq0$ and $\alpha\pm\beta\notin\Phi$ (in this case $x_\beta$ commutes with $x_\alpha$ and $x_{-\alpha}$);
- $\eta_{\alpha\beta} = -\eta_{\beta\alpha}$ if $\langle \alpha,\beta\rangle=\langle\beta,\alpha\rangle=-1$ (the case when $\alpha,\beta$ are the fundamental roots for an $\mathsf{A}_2$ subsystem);
- $\eta_{\alpha\beta}=-1$ if $\langle \alpha,\beta\rangle=0$ and $\alpha\pm\beta\in\Phi$ (the case when $\alpha,\beta$ are two orthogonal short roots of a $\mathsf{C}_2$ subsystem).

When you look at a lift of an arbitrary $w\in W(\Phi)$, the result depends on a particular choice of lifts for $w_\alpha$, which, in turn, depends on the choice of $x_\alpha$.

The numbers $\eta_{\alpha\beta}$ can be expressed in terms of the structure constants $N_{\alpha\beta ij}$ (the expression is usually very simple, but in a few cases involves also the lengths of a certain root chain). It is more complicated in $\mathsf{G}_2$, and in my experience in this case it is much easier to just fix some signs in an explicitly chosen matrix representation and work with that.

The standard reference for all this is "Simple groups of Lie type" by R. W. Carter (Proposition 6.4.3). Since this book is not easy to get legally, you can also take a look at Chevalley groups over commutative rings: I. Elementary calculations by N. Vavilov and E. Plotkin (Section 13).