The GeForce 7600GT will provide you with good performance at medium quality graphics levels. However, it's performance at high quality graphics levels may be too weak for you.
If you have $250 for a video card, then get the X1950 Pro.
Here's a link to an article about Company of Heroes videocard performance. Note the X1950 Pro is not listed.
I have a X800XTPE which can be had on ebay for as little as £90, and it runs the game fantastically. Remember though, RTS's also tax your CPU more than other genres, so poor performance could also be down to that.
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