Yeah, your HD set is only 720p. The higher HD stuff is 1080p which is 1920x1080 resolution, which is higher than the 1280x720 of 720p stuff.
Is it only the game that can't fit entirely on screen, how about normal Windows stuff like web browsing or email? If normal Windows isn't all on the screen, thats the TV's fault.
If thats the case, the nvidia has a screen adjustment thing that you can slightly shrink it to fit on your TV screen. You may be able to do this even if its only the game like that. Since you have the nvidia 8600 I'm pretty sure it can do that, I know most nvidia cards are good about that. Are you using vista or XP?
Also remember, in the Sins settings file there are TWO places where it has a resolution. One is for playing in a window, the other for playing full screen. Its easy to mix these up, I've changed the wrong one myself before.
Example from mine:
VideoFullScreenWidth 1920
VideoFullScreenHeight 1080
VideoWindowedWidth 1900
VideoWindowedHeight 1000
So make sure to change the right one, or just change both.
Try both 800x600 and 1280x720 since you said your monitor supports both.
I'm guessing you can't change the resolution from within the game in the options menu?
Also check to see if this (near the resololution lines):
VideoRefreshRate 60
is at 60, that sometimes helps.
Oh and another thing I thought of. Is your TV set to ZOOM? If your TV is widescreen and is set to zoom normal TV like the news and commercials so it can fill whole screen without bars, it would be doing the same thing to the game

Make sure your TV is set to widescreen 16:9 mode (if its a widescreen set)
Hopefully one of those things help. There can be lots of causes for issues like this.