While you may be able to get your own Blue Card through employment, you may be unable to change your status within Germany. It can depend on conditions attached to your visa.
According to the Federal Office for Migration and Refugees (BAMF) report on
Opportunities to change the residence title and the purpose of stay in Germany
Third-country nationals usually need a residence title if they want to stay in Germany for a longer period of time. The issuance of a specific residence title and the rights and conditions attached to it depend on the purpose of the migrant’s stay. If the purpose of stay changes, the authorities will need to examine whether the residence title is still valid or can be extended, whether another type of residence title might be more appropriate or whether the rights and conditions linked to the residence title need to be changed due to the new purpose of stay. Changes of residence status may also be possible if the person in question has held a residence title for several years and meets certain other criteria to receive another one.
However, this is not to say that you are unable to remain in Germany on your own, with an independent right of residence, and you would need to discuss your options through the authorities at the BAMF.