Adjectives and adverbs: "continuously marked footpath" or "continuous marked footpath"

Question

I have a question about when we should use adjective or adverb in the following sentences. Does it depend on what we want to say or we just can not use one of the following expressions?

Authentically stained glass is colored while the glass is in a liquid state.

Authentic stained glass is colored while the glass is in a liquid state.

In the sentence above can we use "authentically" to describe the adjective "stained"? Or we only can use "authentic" to describe glass.

The Appalachian Trail, extending approximately 2020 miles from Maine to Georgia, is the longest continuously marked footpath in the world.

The Appalachian Trail, extending approximately 2020 miles from Maine to Georgia, is the longest continuous marked footpath in the world.

And the same question about the second example and words "continuously" (continuously marked) and "continuous" (continuous footpath).

Answer 1

Authentically stained glass, sounds awkward to me, but its meaning would be about how the glass was stained:

Authentically stained glass has very strict rules to adhere to.

Authentic stained glass refers to the origin of the stained glass:

Authentic Peruvian stained glass comes from high in the Andes.

The second example can be read (commas inserted for clarity):

1) longest continuously marked footpath
2) longest, continuous, marked footpath

#1 is a footpath that has continuous markings along it, though the path itself may have breaks
#2 is a path that has no breaks, and also has markings on it, but the markings maybe not be as frequently as on a continuously marked footpath