How long does it take for insomnia to wear out after a dexamethasone corticosteroid injection? i had about a 2mg injection and then 1mg, 3 weeks after it. does it take months for side effects to wear out?

From a lot of experience, I can tell you that most people I've seen have a poor night's sleep only for a few days after a large dose of corticosteroids, whether taken by mouth or given by an injection. Of course there are exceptions. In some people the initial sleep deprivation leads to other issues which can lead to a prolonged period of psychogenic insomnia.