I now know the answer, but I wanted to post the somewhat resolution to this problem since there's not a lot of info for troubleshooting burning foot pedals (or at least I couldn't find a lot and it was very frustrating earlier tonight).
I have a Singer Style-O-Matic 328 sewing machine that I inherited from my great aunt who probably got it second hand (since the initials carved into it aren't hers). It's from the early 60s and an amazingly awesome beast of a sewing machine. I work with a lot of heavy upholstery fabric and faux fur and have never had troubles with this machine until about two years ago. Suddenly I couldn't leave my machine plugged in because the foot pedal was making a popping noise and smelled like burning. So I used it sparingly and only plugged it in when I needed it. I didn't take it into a repair shop because I'm really poor and was afraid of how much it would cost to repair.
Today I forgot to unplug it and the darned thing just about caught on fire. Which was quite frightening I must say. I finally took everything apart that I could including the faulty pedal and the transistor part. I read it called something else but can't remember what. On my pedal it looks like a teenaged mutant ninja 9V battery made out of plastic. On the top are two screws that attach to the wire(s) which plug into the wall and on the bottom is a sort of plunger that completes the circuit when the foot pedal is pressed. When I unscrewed everything I found that inside of this transistor thing there is supposed to be a spring that holds the plunger back when the foot pedal is not depressed. My spring had snapped into four pieces causing the plunger to stay put and continually complete the circuit, ergo be on and running all the time.
I plan on going tomorrow to get a new spring and having a romp with a newly fixed sewing machine that should not have gone so long in the condition it did.
I'll update once I get everything all together tomorrow. I'm so excited for being able to figure out what was broken on it finally.