How long does an itchy rash from Amox-Clav typically last after stopping taking the drug?

A few days. Allergic reactions to antibiotics are not common but it may happen. An IgE mediated reaction to a medication may usually present early and with hives and may need interventions such as medications. A late reaction to a medication ( a few days later) that is non-IgE mediated will typically go away on its own once medication is stopped, but it may take another 24-72 hrs to resolve.
If the rash. Is all over your body it might take a good week if the rash is bad. Some rashes need steroids to help clear them up . Are you taking a Zyrtec (cetirizine) every night??