Blind grading for exams isn't very blind
If I understand this correctly, blind grading is a formality introduced by the University to make students' claims on the biased teachers unjustified. I would say - keep it that way and do not bother any more. You are not going to hate them and give them lower grades, are you? So, it does not really matter that you in fact can recognise the handwriting. If a student is really really bothered to keep their identity secret... well, they could try to write with the unusual hand.

Answering your question formally - to implement a real blind grading system you can swap the scripts between departments, or TAs.