Why do I measure low, varying phase leg voltage which fluctuates with load?
RED ALERT! Call the power company TONIGHT and report an outage

You have, at the very least, a lost phase.  And quite probably a lost neutral. 

Now, in most of Europe, this would mean a 100% outage, since their power drops only have 2 wires.  America uses 3, which we get a lot of weird effects when 1 wire breaks. 

The weirdest effect is a lost neutral, where both hot legs are still hot, but the 120V voltage wanders all over the place high and low - the two legs adding up to ~240V, but one higher and one lower.  

The lost neutral is an immediate and imminent threat to your safety.  95% of the time, the problem is on the service drop between your meter and the pole/transformer, since that's the one flapping around in the breeze and accumulating metal fatigue. That's why to call the power company straightaway.  The last time this happened to me, the PoCo was out in an hour, on a Sunday, after saying they'd be delayed due to another emergency. Fixed not only our complex, but a weak hot wire on our neighbor, too. 

The other possibility is a lost hot, which I discuss at length here.  In that case, the other hot will appear to work, but be weird and weak. What's happening is a 240V appliance (like the water heater) has cycled on, and is effectively shorting the two poles through its high resistance heating element.  So your 120V appliances are now in series with the water heater, all trying to share 120V.  That may work well enough for the water heater to reach its high setpoint and cycle off, and that leg will go dead again  Don't do that, though. 

Meanwhile, shut off the main breaker!

You may shut off all breakers and turn the main back on, and then do one of the following: 


turn on ONE 120V load only (no breakers with double width or tied handles) e.g. Furnace or fridge.  It may not work.  
If you know panel topology reasonably well (the weird way that poles are striped across the breakers), you can turn on 120V loads on one pole OR the other, never both... but all 240s must be off.