Yes, Dextron III is ok for your transmission.
But doing the transmission is a little easier said than done. You have to drain the torque convert and the transmission. I don't know what is exactly involved, but it's not too easy. And you must be VERY VERY careful not to get any contaminants (dust, etc) in there, as the trans is very sensitive. It should be changed at 30K mile intervals, no doubt about it.
I would have it changed at the dealer and pay the $120 or whatever it may be. It's worth the money! For any other info, do a search (top left corner) and you'll find plenty of info!
I'm glad you found the site (I think I was the one that referred you?)