This で is a case particle (格助詞), not the one from copula だ.
In those sentences you can treat it as "with".
Not satisfied with simply singing, she also shed tears.
(Actually is a literary/rhetorical way to tell She didn't only sing but also shed tears.)
I feel insufficient with his apology.
I feel as though it could be replaced with "が" in the first example and just left as "は" in the second example.
Both are incorrect. In the second example, replacing it with は makes a small grammatical difference.
彼の弁明では飽き足りない → subject is (usually) 私
彼の弁明は飽き足りない → subject is 彼の弁明 (it makes me feel...; it's felt...)