I don’t need any help, but it was kind of you to offer. = I don’t need any help, and thank you for making so kind an offer.-- No. 'need' is completely wrong.
Poverty is not a shame, but the being ashamed of it. = Poverty is not a shame, but the being ashamed of it is a shame instead.-- OK
Thanks a lot MM. Your remedy to open my another eye. #2, the statement behind is in regard to instead, and the two are in being a shame or not, so the "but" means "instead".