This may not be the obvious change/upgrade to make to your bike when your goal is more speed, but it will work towards that goal: PEDALS!

Have you tried using pedal straps, toe clips or even clipless pedals?

Using any of the three pedal configurations metioned above will give you more pedal stroke efficiency, which means you'll be able to go longer with less energy expended on specific leg muscles. Rather, the same energy required to move your bike is spread out over more leg muscle groups, as you will be able to also PULL UP on the pedal with one leg, while the other is pushing the opposite pedal down.

When I switched to "toe clip" pedals (platform pedal with a cage strapped over it) , and then eventually upgraded to "clipless" pedals (where shoes physically attach to the pedal), I was almost giddy with excitement at how much it improved, and smoothed out my cadence!

As you know... a better, more efficient cadence will equal a faster ride.

PS... you mentioned that you work at Papa Johns... I love that pizza! ...you don't have to make any deliveries on your bike do you!? ha! I'm kidding of course.