Occasionally a sightly grey and blurry vertical line appears near the left center of my screen and sometimes when I play games, like League of Legends, my resolution tears and it seems like the resolution keeps resizing. I think this may be a problem with either my Nvidia GeForce GTX 470 graphics card or my old Dell CRT monitor. I've tried refitting the card in my motherboard, cleaning the card of dust, replacing the vga connector, made sure all connections were secure, reverting to old drivers, and updating to new drivers but nothing seems to get rid of this problem. I'm afraid either my card or my old monitor has gone bad.
my resolution tears and it seems like the resolution keeps resizing
You're going to need to explain what you mean here, because it doesn't make sense.
Usually, though, lines like that are a result of the monitor going bad. CRTs don't hold up that well, and are much more prone to failure than a new-ish video card. I'd be pretty surprised if it weren't the monitor.
Again, what do you mean by widen and shorten? Is the picture becoming compressed horizontally? Because that's not a change in resolution.
Resolution is the measure of the number of pixels displayed on the screen. For this to change on its own unexpectedly would be, well, practically impossible. Differing resolutions will also not change the shape of the picture displayed by your monitor, only the sharpness of it.
It sounds like your monitor is dying to me. A failing CRT monitor can do all sorts of weird stuff, like have apparently random changes in brightness, vertical or horizontal lines on the screen, a picture that appears to vibrate and any number of other things.
I'd recommend ditching it and picking up an LCD monitor. They're pretty cheap these days.