Inside the battery

No, unfortunately your assumption is not correct.

Inside of a battery charges cannot disappear without trace into nowhere. The total quantity of negative charges in both cells is normally constant. But it may be accepted, that negative charges (electrons) move from one cell of the battery into the other cell if both cells are connected with an electric conductor,

Please note too: Charges cannot be separated completely. Thus there are always both positive and negative charges inside the two cells. But in case of a fully charged battery there are far more negative charges.
If the cells of the battery are connected with an electric conductor, the charge distribution will be balanced out.

It is also important that the two cells are separated inside the battery, because otherwise the negative charges would already be transferred inside the battery.

In the next step, you will find out the exact movement of the charges.