A motion to seal/destroy arrest records must be made pursuant to Penal Code section 851.8. These motions can be filed for an adult arrest of PC 484, and if the motion is granted, the arrest records will first be sealed for three years, and then ultimately destroyed. When the motion is granted, the court order to seal the records should be sent to the Department of Justice in Sacramento, so they can then transmit the information to all other law enforcement agencies in California. The motion can be filed even if there was a civil compromise to dismiss, but that doesn't mean the motion will be granted. These types of motions almost always draw opposition from the District Attorney's office, and usually there will be a contested hearing on the matter. You should have an experienced attorney prepare the motion and argue the matter in court. These motions can be filed anytime after the dismissal but must be served on the DA's office at least 10 days in advance of the hearing date.
PC 484 cases can be sealed after a successful 851.8 petition to the court. In order to grant your petition the court must find that there was no reasonable cause to believe the arrestee committed the offense on which the arrest is based. A civil compromise suggests that you admitted guilt by paying the merchant for the damages you caused. Even if the case was dismissed without a finding of guilt a court may decline to issue an 851.8 order where evidence of guilt exists in the court file.
This does not mean you shouldn't try. The only petitions you will certainly lose are those you don't file. You should seek out an attorney in the county where the offense occurred who has had success with these petitions.