Quote:
ruining my reputation among co-workers
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Sounds like you're the one ruining your own reputation.
It is solely your responsibility to find someone to cover your shift. You can't badger someone else into doing it just because your personal conflict "trumps" theirs in your eyes (and in this case it does, but that certainly does not mean they are by any means obligated to help you). You call each of your coworkers yourself, and if you don't get help, then you start calling other stores until you do. If for some reason you still can't get help, then you call your store and say sorry, take care of your daughter, and deal with the consequences later.