Each county is different, but in most counties either the Sheriff's Court Services Division or a Marshal's Division actually execute the levy by going to the bank, etc. They then inventory and take possession of the contents of the safe deposit box. That is why the bank recommended that you talk to the Sheriff's Department. In most counties, the Sheriff holds the goods for a set time, like 30 days, before disbursing the goods or conducting a sale. During that time, you may pay the judgment amount to the Sheriff (not to the creditor) and the contents will eventually be returned to you.
As far as requesting a hearing, you file with the court where the judgment was issued and request an exemption hearing if applicable. If it is a small claims judgment, you may also file a request to pay the judgment in installments. In the latter instance, you will be required to provide your financial informatio to the court and it becomes public reord, so be careful.
Attention: This response is based upon general legal theories and may or may not specifically address issues that affect your individual legal matter or situation. It should not be relied upon outside of the jurisdiction(s) in which the attorney is licensed to practice as each state has different laws. Each situation is fact specific and requires comprehensive legal evaluation following a thorough consultation and review of all the facts and evidence available. This response does not create an attorney-client relationship between the asking and answering parties.