We just had new kitchen sink installed and now when we run water both sinks fill up and empty slowly. On the old sink we had an air vent for the dishwasher but the plumber said that is not code anymore? Can you help me?

Please explain what you mean by a air vent for the dishwasher? Do you mean a air gap for the dishwasher, usually a chrome covered device on the sink rim, or do you mean an in line vent on the pipeing underneath the sink?

What did the plumber do to bring it up to code. Generally you do not have to do anything to bring something to the present code that was approved under a previous code.