Elantra Shaking Badly, CEL on, Dealer says it is misfiring

Hey everyone, I'm kinda stuck here.  I have a 2013 Hyundai Elantra GLS PZEV with
less than 10,000 miles on it.  While driving on the highway, I faintly smelled something
weird with the heat on.  The check engine light went on and the vehicle started shaking
badly, from the engine compartment.  While coasting the vehicle was fine and the
check engine light would turn off, but the moment you gave it the gas, regardless of
speed, the check engine light would pop back on and it would start shaking violently
again.  Over a short distance, the speed dropped a bit, eventually resulting in much
higher engine RPMS just to stay at 60mph.  I quickly pulled off the highway, and had it
towed to the dealership.  When the car idles after the shaking, it still shakes
sometimes with the check engine light on, and the lights of the car dim off and on.  If
you turn the car off and turn it back on, it runs smoothly, until putting it in drive.

Dealership says that it is misfiring without saying what cylinders or how many
cylinders are misfiring.  They also say that it is because of bad fuel and it won't be
covered by the warranty.  With a couple exceptions (but none recently), I have always
gotten my regular fuel from the same place, Citgo, as they sell the most gas in town
and usually should have the cleanest stuff because of that.

The dealer is trying to sell me on a fuel induction service for $130, and also saying
that the entire gas tank should be drained (when it is a full tank), which would be crazy
expensive unless I did it myself.  I'm not entirely sure how allegedly bad fuel could do
something like that, especially when I've been driving a lot lately so I've been going
through gas a lot quicker.  Is a higher octane fuel really needed for an elantra?

Any idea what could be causing my problem?  Thanks