A previous landlord filed a Garnishment, faxed the Garnishment paperwork to my husband's place of employment on a day that he wasn't working, and we had already paid the debt in full one day previous to finding out that he had filed the paperwork.
What legal grounds do we have for the sending of the paperwork via fax for all to see and read and where do we stand legally with this filing? Since he cashed the check, doesn't he have to withdraw the Garnishment?

Absolutely. He cannot garnish your wages for a debt that is not owed.

What about the court fees and filing fees that he incurred by filing this paperwork? And the original debt was to be paid by a certain date and wasn't; although we had been sending lesser amounts each week until we could pay the debt and he was takingthose payments as well. On the deadline date, he filed for the Garnishment, still taking the smaller payments through these last few weeks, as well as the final large payment ( he went straight to the bank with that one!)
My question is; since we did go past the court appointed date for final payment, are we still liable for any further payment to him, including filing fees?

If the landlord obtained a judgement, then the judgement was for a specified amount of money that might have included court costs and interest. If the amount that you paid was less than the judgement awarded, then the garnishment can still stand.

You need to show your husband's employer how much of the judgement has been paid so that they deduct only the balance.

No, we paid the full amount that was stated in the judgement.
The difference would be the court filing fees that the landlord incurred by filing the Garnishment.

No, Court fees are not owed unless awarded as part of the judgement.