Well, I finally got the new TV working by plugging the old monitor back in and then following the procedure from the link in my OP to tell windows it's a non-PNP flat screen monitor. Obviously this wouldn't have worked if my old monitor was broken, rather than just being changed out. I can't believe that with all that has happened in the computer industry, ATI doesn't have a way to let you use new monitor when your old one breaks, but judging from my lack of success in both my own posts and looking at others I found on Google, it really doesn't. I saw a few guys with the same problem who reported that NVidia cards worked could handle this, but ATI cards don't. It seems I would have had to reinstall Windows in order to get ATI to work with my TV.
Anyway, I now have a great picture at 1680x1050 and several smaller resolutions, but I thought I should be able to get 1920x1080. THAT problem is probably specific to my TV, which is a new model, so I won't bother people here with it.
Thanks to everyone who responded; I appreciate your efforts.