Dsiee, thanks for the help.
You can connect your external tuner (assuming that is what your converter box is) to your new TV without a great risk of damage (no greater than connecting DVD player).
The converter box is what I bought to convert HD signals to work on my CRT. I'm assuming that's the same thing as a tuner.
You may have an option to turn on analogue channels on your new TV which may allow you to get the channels you are missing out on (Assuming analogue TV is still being broadcast where you are).
When I set up my flatscreen, I did see the tv was scanning for HD and analog channels. At the end of the scan, it showed it only found HD channels.
I was able to improve both situations with a little more fiddling. The preset picture settings are horrible. There's only one user setting which allows me to adjust brightness, color, etc. The optimal setting for HD channels doesn't make the non-HD look very good. And vice-versa. So I've compromised and set the picture to work with both. M favorite channel is one of the non-HD channels.
I moved the antenna to another area--a place I'm not crazy about, but it seemed to be the place that worked for the channels I use the most. Still not as good as the converter box, but an improvement.
I'll try to connect my converter box to the tv and see if the reception improves. I had offered my old tv combo to a friend--may have to rescind the converter part of it.