There are two possible genders in plural: masculine personal and everything else. 'Nasi' is the masculine personal form, meaning it can only be applied to masculine nouns which describe people (e.g. 'nasi chłopcy' - our boys, 'nasi ludzie' - our people - even though the group of people may contain women, 'człowiek' is a masculine noun).
Everything else goes with 'nasze', e.g. 'nasze psy' - our dogs (even though 'pies' is masculine it is not personal), 'nasze dziewczyny' - our girls, 'nasze krzesła' - our chairs etc.
It's not z gone, it's sz changing to ś. (s before "i" always sounds like "s"). Those are very different sounds in Polish, but learners tend to have trouble with hearing the difference.
nominative plural masculine personal of adjectives softens the last consonant (if possible). the same thing happens with "nasz, wasz". Sometimes those shifts are more "visible" "brzydKi-brzydCy (ugly), drogGi-droDZy" (expensive/dear). and sometimes they are only "audible" "ładny-ładni" (n before i is "ń")