Everything that uses the Color mode isn't flashing, but as soon as I use Texture, it goes berserk.

What minecraft version are you using? And whats your mod list?

i am assuming this is the latest version, because i have the same issue on 1.20.1 fabric (all mods are updated). for me it seems to only flicker when it is rendered on top of other parts (if the glowing part is half in front of the head and half not, the part over the head flickers constantly)

Add 0.01 to the MCScale of the glowing part, to prevent z-fighting.