What is high potential energy?
What? Electrons are propelled by F = qE = qV/d where V is the cathode voltage across a gap d to the anode and q is the electron charge. But up against the negative voltage, the E source, is where the high, not low, potential is. How do we know this? Because that force F will work on the charge over the distance d and cause it to gain kinetic energy as it crosses d from the cathode towards the anode. That is, the potential energy PE near the cathode is converted into kinetic energy by the work the force F does on the electron q. The potential, stored energy is highest close to the cathode, the negative voltage. That is PE is transformed into KE by the push from the cathode voltage and the E field it creates. So the electron is moved from a higher potential energy PE and lower ke, to a lower pe but with higher KE. Higher to lower PE as one would expect when work is done and not "from lower to higher potential" as you assert. Same thing for the protons. They are accelerated from the anode side of the E field and drawn to the cathode end. So they accelerate, gain KE, rushing toward the cathode side of the E field. So, like the electrons, but from the opposite side because they are positive, not negative charges, the protons are at their highest PE and lowest ke near the anode, and rush to the lowest pe and highest KE near the cathod…higher to lower PE just like the electrons. Think of the gravitational PE = mgH at some height H off the surface. It's traded in for kinetic energy as an object with that stored energy is attracted and accelerates towards the source of the gravity, Earth's mass. Its PE is highest farther away from the source of the attraction. Same deal with electrons. They are attracted by the anode; so their PE is highest when farthest away from the anode, i.e., near the cathode. The clue is in knowing which way the particles go and increase their speeds and, consequently, their kinetic energy. As the kinetic energy goes from ke to KE > ke in a given direction through an E field, the potential energy must go from PE down to pe < PE as that kinetic energy increases because of the conservation of energy law. That is, the total energy remains the same; so as the KE increases the PE must decrease in compensation.
Increases. It's like rolling a ball uphill against the force of gravity…the negatively-charged electron naturally wants to move from low potential (more negative, so the electron is repelled) to high potential (more positive, so the electron is attracted), just like the ball naturally wants to roll downhill. So the potential energy of the electron increases as it moves to a lower potential, because work must be done on it and that work is converted into energy.
Think of it like a rock at the top of the cliff. The top of the cliff is high PE, whereas the ground is lower PE. As the rock falls, it is decreasing in PE (gaining KE).
Same deal for electrons. As they go from high to low potential, their electrical potential E is decreasing as its converted into KE and whatever else energy there is.