Anyway and However are both adverbs, but perhaps having different function with the sentences they are used to.
First,
HOWEVER has a core meaning which is an adverb introducing some form of contrast.
For example: The letters are stained and faded. I do, however, believe they're legible.
and
ANYWAY has a core meaning which is an adverb meaning no matter what the situation is.
For example: Anyway, we have to pay whether it was accidental or not.
Recycling, according to some anyway, is the best way of teaching respect for the environment.
I hope it could help you in choosing the best answer. Thank you.