If an effect allows you to cast a spell without paying its mana cost, you can’t choose to cast it and pay unless another rule or effect allows you to cast that spell for a cost. Similarly, you can’t waive a cost reduction unless that effect says you may.
Adamant effects check what mana was actually spent to cast a spell. If an effect allows you to spend mana “as though it were mana” of any color or type, that allows you to spend mana you couldn’t otherwise spend, but it doesn’t change what mana you spent to cast the spell.
If you copy a spell that has an adamant ability, no mana was spent to cast the copy at all, so that ability won’t apply.