I think your version of Windows might just be too old, sorry.
If you have a little space on your drive, you can
try out new versions of Windows for free - Microsoft allows you to use trial versions of fully functional versions with very little restriction for 90 days. You just need software that will create a dedicated partition for it so that you do not overwrite your old Windows install. I would not suggest using these as an upgrade (in fact I think that's disabled for trials) until you get a legit key - and a Home or Pro key will work just fine.
The reason why I suggest this is there's no reason your hardware can't run it - but I know Graf has been upgrading his compiler over the years and as he keeps using different ones, and as they upgrade they silently drop support for earlier versions of Windows. Frankly, I am surprised GZDoom even ran on Windows XP.
Since only Windows 8.1 and 10 is listed, I would also suggest getting
Classic Shell - this will give you back an actual start menu, one that you can even customize to be a little more like Windows XP if you want.
I would not suggest trying this on your laptop; at least not right away - only the newer computer with the 3GB of RAM. As much press as Windows 8 gets for being lean and efficient, it's still a slog compared to XP.