I used Curseforge to download this mod for Fabric 1.21.4 after seeing it online where it was showcased with dirt overlay on blocks and some other things including copper block and I think prismarine block connected textures.

You need a resource pack containing said features for that.

Continuity does not include stuff other than what you mentioned working.

Continuity just adds the functionality.

That is correct. Try searching for connected textures resource packs on Modrinth, CurseForge, or Planet Minecraft.