One apology --> many apologies.
One good deed --> many good deeds.
One sorry --> many _ ?
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"Sorry me no sorrys, my Lord Duke," cried the Duchess
(I have a feeling that this is slang employed literary, so it is not necessarily grammatical use)
Junʼichirō Tanizaki in his "A cat, a man, and two women: stories" gets translated into
Then, with twenty or so thank-yous and awfully-sorrys, he went on to ask for the loan of a lamp for his bicycle...
The word is definitively used in spoken language (most examples analyzed are such examples). It refers to the multiple acts of apologizing, making a noun out of the adjective, according to following logic:
Your sorry = Your apology
Your apologies = Your sorries/sorrys
The spelling is not a problem in spoken language, but in writing it is. I suggest to quote the word to indicate unusual usage.