To answer your question: Yes, you'll definitely need a new rig. I'm surprised that you can play the current version with the one you're using now!
As for the dxdiag report, instead of a screen image use the "save all information" button and send the DxDiag.txt file. But I must give you kudos for having run the DxDiag and posting the most essential parts of it. That helps to explain you the bottlenecks in layman's terms.
As you may know, every game (and any other program for that matter) has minimum requirements for the hardware. The current game leans more on the processor side than on the video card side and that's why you've been able to play it at all. The alpha puts a lot more struggle on the video card like most games do. Your video card was the weakest of its series already in 2007, it barely meets
the current minimums. Truth to be said, if that computer was sold to you as "a serious gaming rig" you have been fooled.
As for you not being able to install to Win7 or 8, did you try to do a clean install or an upgrade? An upgrade install over the existing one is never a good option, because it carries all potential problems to the new system as well. There may be driver issues and other settings related things, not to mention the possibility of malware.
One obscurity I noticed in your DxDiag is that you seem to have 6 GB of RAM in a 32 bit operating system. Shortly put, no 32 bit operating system can use more than 4 GB of memory, so the rest is just sitting there. With a 64 bit OS the limit with current components is measured in
thousands of miles rather than gigabytes.
Summa summarum, for your current rig to be able to run the Alpha, you'd need a better video card. But then you'd probably have to get a more powerful power supply, too. Those two alone can be a good start for a piece-by-piece build, but for longevity I'd recommend getting everything needed at the same time.
BTW, it might be interesting to know the specs of your other system to see if combining the two provided a better gaming experience. A Dr. Frankenstein approach to computers...