I'd like to apologize for not being able to do xyz.
Rather than just saying 「すみません、何々ができません。」, could I say something like:
Or is しまう always used with a positive te-form verb?
Firstly, I'm not very knowledgeable about grammar so I might be talking about a different usage of the same しまう.
しまう is used when you have done something unintentionally or unwillingly. For example, from the sentence:
you can tell that the person didn't break the vase on purpose, whereas in
you can't really tell the person's intention.
Now for your example, できなくてしまいます is できない + しまう. できる/できない is about your capability of doing something. When you're capable of something, you intentionally do whatever you're capable of. Therefore, it doesn't make much sense to talk about something you can't do, but can do at the same time.
That being said, できてしまう is used, I think mostly in novels, when you want to say that you are capable of doing something that you are not willing to do. For example,
means that you are able to give up, even though you are unwilling to do so.
It seems to me that you were trying to force the English word order into Japanese, the got into trouble with attaching しまう to negative verb forms. In English you say "I'm sorry that I was unable to help". You apologize first and follow up with the explanation. In Japanese, the word order is the opposite. You say "助けてあげられなくて、すみませんでした". In some sense you can take this literally as "I failed to help, and as a result something was left unfulfilled". It preserves the temporal order of the two events, and there is no need to fuss with しまう.
しないでしまう is not common (marginally grammatical).
しなくてしまう is extremely rare.