Your old 15" is a standard definiton TV, and yes, it will look better on a standard definiton box. The set you purchased is a high defintion TV, as are all the flat screens currently being sold.
The bigger the screen, the worse it will look on standard definiton signals. Coax and composite (red white and yellow) will give you the lowest possible resolutions as well. If your box actually has the component (red,blue,green) it will be slightly better, but you will still see artifacting/blurring/shadows. The sharpness settings on the TV will do nothing. The only true way to improve sharpness is having an ISF cailbration done, but you need a high def source, preferably bluray, to acheive the proper settings and best picture.
I highly suggest you get a HD box and hook it up via HDMI. If the issue is still there, then the TV is NFG and you have 30 days to exchange it, provided the TV is in new condition and you have all the accessories and original receipt.
