"Is there a Newtonian or some other explanation for dark energy?"
Dark Energy is by definition a Newtonian "band-aid" by formulation and name, to describe Universal expansion.
"It seems there was insufficient dark energy in the earliest moments of the universe to accelerate its expansion; even if briefly the universe went through an exponential inflationary phase."
But we cannot know how much there was prior to the CMBR quench, in any way, shape or form. And Dark Energy is no sort of "stuff", because it is distributed spatially *perfectly* at every epoch.
"And so, early on if it wasn't accelerating after inflation, from where does the force come to cause it to accelerate at some later time."
Precisely the problem with the entire concept as presented.
Compare the inflation, stagnation, acceleration of expansion curve, to the "bathtub curve" of a product's life. Inflation then corresponds to initial sub-component failures. Stagnation corresponds to normal service life. Then acceleration of expansion corresponds to individual sub-component failures due to fatigue at the end of system service life.
Or better still, inflation is production of interior spacetime from an exterior spatial radial coordinate, stagnation is the fall towards the central singularity, and acceleration of expansion is mergence with the central singularity:
http://math.ucr.edu/home/baez/gr/oz1.html
... just follow the yellow brick road.
"In my simple way, I cannot imagine that gravity would lose its grip as it condenses matter into galaxies, to then be replaced by another force we call dark energy."
But here is the thing with that argument: if you are inside a planet (Newton's shell theory), the mass outside your shell has no net pull on you. If the Universe is tiny, then there is just as much mass to your "left", as to your "right". There is no particular reason "on average" for you to be "slowed down" or pulled in any particular direction.
"Naively, nor can I imagine there is a 'relativity' or even 'quantum' explanation that completely rules out 'force = mass x acceleration'."
Except that gravitation is NOT a force in Relativity, nor in Nature, and mass is not a fundamental property in Quantum Mechanics. You have to give up grade school explanations / models, if you find these issues that arise from them to be too uncomfortable.
"Nor can I imagine that the energy of space itself is being continually increased out of nowhere. If someone has an explanation, I would be delighted to hear it."
Dark Energy is just the shape of spacetime, variable in TIME not over space.
I know this is uncomfortable, and I hope you continue to use your head... you have put your finger one some real problems. And you can ignore them, and live a happy life as yet another zombie flatlander...