Styg_lan, I had the same problem initially. In the Stardock email I receieved, it provided a download link. I downloaded the exe to my desktop, ran it, and it installed (apparently) Stardock Central a second time (even though I already had it, and indeed, it was what I used to navigate to the page I purchased the download on.) Anyhow, once I did that, it still only said "Purchase," but I closed Stardock Central and reopened it, after which it said "Install" next to it, and I was able to download and install Dark Avatar. I hope that helps.
To any Stardock devs reading this: it's everything I hoped it would be!