It is possible to change from a student visa to a work visa, even if you drop out of university. The German government changed the regulations on such cases as part of a law it passed called the Fachkräfteeinwanderungsgesetz in 2020.
Please bear in mind that a work visa can only be issued for job that require a qualification, generally a university degree or formal vocational training (Ausbildung). If you have a foreign qualification, it may need to be recognised. For some nationalities there are exceptions to this rule.