Sorry, I must've missed this previously...correct, we only hop on Verizon network when roaming, but if you're using the 00001.prl, it won't show you as roaming on your phone, but both Sprint and Verizon will know you're roaming. What I mean is the icon for roaming (little triangle) doesn't pop up. Using the 00001.prl is basically like having a Verizon account...you're on VZW network 100% of the time, their 3G speeds, their spectrums (better building penetration), etc. 100% of the minutes and data you use will show up as roaming on your bill and if you abuse it, you'll be dropped. I loathe the people who do this intentionally to get out of an ETF, but that's my opinion. You signed a contract and had 14 days to test it out....but that's not what this post is for.
Using the corporate PRL (24006 I think) enables 3G roaming speeds, where as using a 'standard' PRL (25006) only has 1x roaming speeds.
As of right now, there is no way to force roaming on this phone. I'm sure it'll be figured out, but some of the best devs are having some problems finding exactly what it is. The only way to 'force roam' is basically to put the 00001.prl on your phone using the described methods above. Anycut is a GREAT tool for this.
You spell out data with numbers, you can't actually get letters in there. So you're right, it's technically ##3282#. If it's not working, then that apk (I think it was called HiddenMenu.apk in the OG Epic, not sure about the GSIII) isn't present and that command won't work. Get to an actual TW ROM and you'll jump right into it.
My assumption would be if you can't get there through the dialer, using the Anycut shortcut won't work either.