Can I be evicted if I paid my back rent in full?

You should contact the attorney and fax proof of payment if possible. You need to go to court to avoid a default judgment against you. Especially after you have gotten caught up on the rent payments.

Dear New York Tenant: You may have the papers in hand but may be misreading. Likely, the five day notice you received was a "Rent Demand." A rent demand is a legal document that comes before the landlord may initiate a nonpayment summary proceeding. The rent demand should say that if the rent is not paid within five days and if not paid you do move out, that the landlord will initiate a summary proceeding. The summary proceeding in the nonpayment case is based on the same rent demand. The landlord requests a judgment for possession based on the claim that the rent was not paid even though the landlord made a demand for payment. The notice of petition tells the tenant that the landlord is seeking a judgment for possession based on the claim of nonpayment of rent. It must inform the tenant to Answer the petition to avoid a judgment. When a tenant has paid all the rent claimed owed by the time the landlord serves the notice of petition and petition, the tenant has a complete defense to the lawsuit based on the fact of paying the rent, but no one will know about this defense if the tenant does not answer the petition. A notice of petition in a NYC Housing Court nonpayment case does not have a court date. The tenant must answer the petition to gain a court date. Read more at: https://www.nycourts.gov/COURTS/nyc/housing/answering.shtml

It appears that a default judgment was entered and that the judgment should be vacated. Go to Court, ask for an Order to Show Cause staying (freezing) the eviction and scheduling a hearing to vacate the judgment. Only the Court can vacate the judgment.