Your text is being cut off, as you probably have figured out, because offset looks at what is seen on the screen at moves that. So your text that is outside of the screen can't be offset as it's not seen by the effect.
Probably the easiest solution would be to create your text in a precomp, offset it there and bring it into your final comp to work as a scroll. Click your text layer and "Cntl + shift + c" to create a precomp. Delete the original offset. Open your precomp and size the comp(cntl + K) to fit your text layer(if your text is 1200 pixels wide and 100 pixels high, make make your comp 1210 x 110). Now apply your offset effect and animate it.
Here's a quick example project to illustrate what I hoped I explained:
4030_offsetingexample.aep.zip
Johnny Cuevas, Editor
Thinkck.com
"I have not failed 700 times. I have succeeded in proving that those 700 ways will not work. When I have eliminated the ways that will not work, I will find the way that will work."
---THOMAS EDISON on inventing the light bulb.