Problem showing the saturation map of a graded ring f.g. in degree 1 induces an isomorphism of projective schemes and O(1) (The Rising Sea 15.6.G)

Question

I've had problem working on this exercise from the July 31, 2023 version of Vakil's The Rising Sea:

15.6.G. Exercise. Show that the map of graded rings $S_\bullet \to \Gamma_\bullet\widetilde{S_\bullet}$ induces an isomorphism $\operatorname{Proj} \Gamma_\bullet\widetilde{S_\bullet} \cong \operatorname{Proj} S_\bullet$, and under this isomorphism, the respective $\mathscr{O}(1)$'s are identified.

It could be easily seen that for homogeneous $f \in S_n (n>0)$, we have isomorphisms on $D_+(f)$ in these two projective schemes that glue nicely. However, I could not prove $\Gamma_\bullet\widetilde{S_\bullet}$ is finitely generated in degree 1, thus I could not show $\mathscr{O}(1)$ on $\operatorname{Proj} \Gamma_\bullet\widetilde{S_\bullet}$ is an invertible sheaf. Nor could I prove this map of graded rings induces a map of projective schemes defined on all of $\operatorname{Proj} \Gamma_\bullet\widetilde{S_\bullet}$, since a general map of graded rings only induce a morphism on an open subscheme. How could I prove these two statements?