1. ὠς σωφρονέστατα: I know it looks like an adjective, JM, but that's just the way superlative adverbs are formed in Greek — by using the neuter plural of the superlative adjective. Alone, σωφρονέστατα means "most prudently." With ὠς, it means "as prudently as possible." You can do that with ὠς (or ὅτι) + any superlative adjective or adverb to mean "as ... as possible." "As beautiful as possible," for example: ὠς καλλίστη, or ὅτι καλλίστη.
So the first part of the sentence means "We shall act as prudently as possible."
2. Something is wrong with the second part of the sentence, because "until we feel ashamed before at least (γε) the better men" doesn't make logical sense, does it? If you're acting prudently, you should not feel ashamed, right? Something tells me there's a misprint here, maybe a missing μή (or οὐκ, if Hansen wasn't on his grammatical toes!).