I had an HD5770 (it's actually the same as 6770) for some time, and I still have it now. It really wasn't a bad card, but as for modern gaming... Yeah, it's lagging behind. As for Freespace, I remember that it was doing fairly well, but shadows on a larger amount of objects (like asteroid fields, but bigger battles too) were out of the question.
That card has a max power draw of 108W, so it would only be a problem with a really cheap chinese PSU. And well, in my regions you can find used ones for an equivalent of 25 USD.
Perhaps the best way to get a better PC is to actually upgrade the one you've already got? I'd suggest you to post your current specs here, maybe someone might come up with something. Actually, show that prebuilt PC you're considering too. And maybe tell us your budget for the upgrade? You might be getting a lot more from that money than you anticipated.
And "my W7 machine is getting slower" - slower meaning "With all this modern software and games, my hardware just isn't keeping up" or "My Windows 7 wasn't been reinstalled for some time now and it's getting really frustrating to work with"?
Edit: Forgot to mention here, some motherboards in prebuilt PCs (usually Dell and HP, but there can be many others) may come with a PCIE x16 slot with severely limited power availability, i.e. some Dell boards limit their PCIE power to 25W out of typical 75W, so you may want to be careful.