The 22-year-old former Accrington man has joined after Dave Richards broke a finger in a reserve match.
Dawber has had spells with Fulham, Aldershot and Bishop's Stortford and spent the first half of the season with National League North side Altrincham.
He has trained with David Artell's side in recent weeks and is set to be on the bench against Newport on Saturday.
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League Two side Crewe Alexandra have signed goalkeeper Andrew Dawber on a deal until the end of the season.