I already hold H-1B status. Can I switch my job and still keep my current H-1B visa?

No. Because the H-1B visa is employer specific, you may not be employed by a new employer until the new employer files a new H-1B petition on your behalf. Because you are already in H-1B status, you do not need to wait for the new H-1B petition to be approved and can start work as soon the new employer receives a fee receipt from USCIS stating that USCIS has accepted the H-1B petition for processing.