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It is not possible to apply for any US visa whilst you are in the US.
All applications must be made in person at a US embassy/consulate outside of the US, so even having a second passport is not going to help you here. Further, your new status only starts once you re-enter the country using the new visa.
Under certain circumstances it is possible to apply for a "Change of Status" whilst you are in the US, however I would be fairly certain that a change from J1 to F1 would not be allowed.