To understand what's actually happening, first we need a clear picture of the basic physics. For example, electrons themselves aren't stable/unstable. Instead, each conductor as a whole can develop a charge-imbalance.
We could say that a conductor as a whole is electrically "stable" or "unstable." But the conventional terms for these are "neutral conductor" and "charged conductor."
A charged conductor can attract or repel the mobile charges found inside other conductors. (These charges aren't always electrons. For example, the mobile charges inside battery acid are mostly +H ions, also called "protons.)
All batteries are charge-pumps. In a battery, the chemical reactions force some charges to move across the conductive electrolyte between the plates. As a result, one plate becomes electrically charged positive, and the other charged negative.
This "charge pumping" action doesn't continue forever. It only operates until a particular value of voltage appears between the battery plates. We can think of batteries as being "self-charging capacitors." Their internal charge-pump will only run until the "capacitor" is charged up to a particular voltage (such as 2.0V for lead-chemistry batteries.) Then the charge-pump shuts down.
Now to your question. If we have two separate lead-acid cells, and their charge-pumps have run just enough to put 2.0VDC across their terminals ...then what happens when we connect them in series? Ah, that depends on the total net charge of each cell, which is a very different concept from the charge found on each battery terminal.
For example, if all the parts of the battery were electrically neutral before manufacture, then when we put the battery together for the first time, one plate charges up to +1.0V, and the other plate charges up to -1.0V. Together, the net charge of both plates is zero. Compared to the nearby floor, the battery as a whole is uncharged. The two plates cancel out. But each plate considered alone is certainly charged!
Next, take two of these "electrically neutral" cells, and connect them in series.
At the moment when the +1.0V terminal of one cell touches the -1.0V terminal of the other cell, there will be a tiny spark. The terminals of the two separate cells will act like the two plates of a capacitor; a capacitor which has just been shorted out. A brief current appears as they discharge each other. Their potential-difference goes to zero.
But now the charge-pumps within each cell will see the cell-voltage drop by half! The 2.0V across the cell has fallen to 1.0V (the potential of the second, floating terminal.) The charge-pumps in both cells see this decrease, so both turn on. They pump charges through their electrolytes until the voltage across the electrolyte rises again to 2.0V.
As this happens, one charge-pump is pushing charges into the common terminal between the two cells, while the other charge-pump is taking them out again. Compared to distant ground, the potential on the common terminal between the cells is remaining at zero.
But look at the other terminals. The charge-pumps run, and after a few microseconds the potential of one battery plate will grow to +2.0V, while the potential of the other grows to -2.0V. Finally, we have 4V difference between the floating terminals of the two cells in series. The charge-pumps then shut down.
All of the above is assuming that the two cells were electrically neutral before we started. If instead one of the cells started out with an overall 1.0V charge with respect to the earth, then one of its terminals would be at 2.0V, and the other at 0.0V (giving an overall 1.0V average for the entire cell as a whole.) Connecting it in series with another cell would give a different result initially. But then the charge-pumps would run, so that each cell again had 2.0V between its terminals.
The above is a physicists viewpoint. In electronics, usually we draw a ground symbol in our circuit diagrams, then declare that the grounded section is electrically neutral, with zero volts wrt earth. This can simplify things. Try drawing a ground symbol in your schematics, then explain things while maintaining one section of the circuit at constant zero volts.