Okay, this has taken a l-o-n-g time to reach this point (this is not the first time I've tried to achieve this). But finally I have come up with an approach that I think works well using Vegas' "Scrolling on Transparent" text generator. There is the option to change the scroll direction from "up" to "down" (which removed the need for key frames).
It turns out to be a matter of timing and placement. I was trying to start them at the same same, which was a mistake. The timing is: 6-sec clip, 8-sec clip, 10-sec clip and aligning them flush on the right-hand side.
Next, alternating the text placement from left-center-right and then rotating that placement every other grouping gave me the desired effect: L-C-R, C-R-L, R-L-C, then back to L-C-R. Once that group is done, it can be copied and pasted as many time as the project requires. Granted, you have to go in and change the text for each, but that's no big deal.
You can see a video of the test here.
Anyone who would like a copy of the veggie (V9.0), drop me an e-mail and I'll send it to you.
Thanks to all for your help!
EDIT: Looks like this on the TL: