Can you use -なくて instead of -ないで in the following sentence, keeping the same meaning? What's the difference between ないで and なくて ?
Peter Sells (1995) calls ないで as verbal gerund and なくて as adjectival gerund. When you have participial constructions, they do not make difference, but Sells notices that only the verbal gerund can be selected by an auxiliary verb:
Similarly to that, when you want to use these forms adverbially as in your examples, you have to use ないで. You cannot replace it with なくて. If you do so, they would be interpreted as participial constructions.
Other than that, I do not know of a difference.
Sells, Peter (1995) "Korean and Japanese Morphology from a Lexical Perspective" Linguistic Inquiry 26.2 MIT Press