Using Continuity without Sodium works just fine but when I add Sodium the connected textures stop working.

You must use indium alongside sodium

https://github.com/IMS212/Indium/actions/runs/2458463120 you can find a temporary solution until an official update is released.

I have confirmed the fork worked for me

I was looking at indium but it's only for 1.18.2 currently and I'm on 1.19. Does that version still work in 1.19?

No, use the linked build.

How do I download it using the linked build?

@JacobWasTaken

and then

Ok, thank you!