An adverb changes meaning, such as establishing where, when, how something happened and to what degree.
For example: Rosie sat at the back of the car.
In this case, 'back' reveals where Rosie sat.

A preposition connects 2 pronouns/nouns (he, she, john etc.)
For example: I put an apple into the fridge
'into' connects apple and fridge (nouns)

So 'across' can be both because
'I walked across the road' establishes where I walked to and also connects the pronoun 'I' and the noun 'road'