120V on hot and neutral wires… just started happening
Current flows in loops

Remember, current flows in loops. It's not like air, where you have the "hot" compressed air lines coming off the compressor, and there is no air return, it just dumps air wherever.  With electricity there must be 2 wires to complete the loop, and both are equally important!  We make all the fuss about the hot wire, but neutral is an equal partner in the deal.  Just an often-overlooked one! 

So if your hot wire had broken, you'd have sussed that out in a minute.  Now you must "think upside-down" and look for the neutral wire break. 

Part of your confusion is "phantom voltage". An isolated wire (disconnected at both ends) will pick up stray electrons due to capacitive coupling from hot wires nearby (i.e. in the same cable).  A hi-impedance (any cheap digital) voltmeter will see those electrons as an actual measurable voltage, even though they're not.  You've been spending way too many brain cells chasing that phantom, so let's nip it in the bud by removing the high impedance.

Kill the phantom

Start by plugging a load into a dead outlet and turn it on - preferably something tiny; an incandescent night-light is perfect.  This will connect hot to neutral (through the medium resistance of the load), and if neutral is disconnected, that will instantly yank it up to 120V. Goodbye phantom voltage.  You were seeing that earlier, until you unplugged your (I'll guess) toothbrush charger.  Plug something back in.  This will suppress phantom voltage. 

Now you can troubleshoot upside-down: if you measure neutral and it rings out "hot", that means you're afield of the wire break.  If you measure neutral and it rings out 0V, that means you're on the panel side of the wire break. 

I suspect the problem may be -- are you sitting down? -- at the neutral bar of the panel. 

Speaking of "Panel"

That is the dreaded Rule of Six panel.  It also looks VERY full. All such panels are a serious hazard, because as you may notice, the 6 breakers sum up to way more than your service, and there's no master disconnect. 

Other than that, Mrs. Lincoln, it's Square D QO, which is absolutely top shelf. Great panel. If only there was a way to keep it! 

A year ago, I would have walked you through converting it to a "Rule of One" :) panel by adding a subpanel.  But then, NEC 2020 dropped.  NEC 2020 requires an outside disconnect switch on every new service, reasonably near the meter.  So now, my advice is to wait and see what this requirement does to equipment prices, and then, as soon as feasible, either add an outside disconnect which is also a breaker at service amperage (e.g. 100A if that's your service)... or replace your meter box with a meter-main which combines both.  That permanently cures the problem with Rule of Six panels, and it may continue in service forever with no concern.  And your QO panel is very high quality stuff, so you keep benefiting from that.