@bmacDonald94 Did you try the apk from their website?
https://signal.org/android/apk/
The store version uses google cloud messaging to notify when there is a new message. Website version doesn't have this dependency, however it is polling based so will be harder on the battery life.
In both cases, Signal does still use Cloudflare for (encrypted) message passing.
Web apk can't be updated via stores, only by that link; but can confirm, does work without google anythings (play services etc.)