My overall goal is to use an arduino uno to control a servo to rotate continuously in order to drive my robot in a straight line.
This requires modified servos. What kind do you have?
First of all, when I use the regular, unedited Sweep code with my servo, it rotates one nearly full rotation, stops, then rotates again in the same direction.
Why are you running code for regular servos with your modified servos?
My main problem is the motor stopping. I tried taking the delay out, or setting it to 0, and it still did not rotate continuously.
So, don't tell it to stop. Continuous rotation servos (which really aren't servos) are driven using the writeMicroseconds() method, not the write() method. The values range from somewhere around 600 for maximum speed in one direction to around 2400 for maximum speed in the other direction. In between, the values result in slower speeds, until some value (or small range of values) causes the "servo" to stop. You'll need to experiment to find the values that represent max speed each way and stopped.
The "servo" I'm using is actually a Vex 2 wire motor with a 3 wire motor controller 29. The motor controller 29 is supposed to make it able to be programmed as if it were a servo.
Isn't there any documentation you can refer to (or link to so we can read)?