Your list of conditions is rather restricting...
The Intel chipset is going to limit you more than anything else. It doesn't matter how much RAM the chip is getting if the bells & whistles the game wants are not present. The system RAM is a bit low for a dual core, especially for sharing with the integrated graphics. (consider capping the VRAM share in the BIOS, 512 or 256, depending on what you're doing)
How much free space do you really have, and what's filling up the HDD??? If you've got an external drive, consider offloading as much as you can. Some game examples: Company of Heroes (One) was new in 2006, needed a Pentium 4 / 512mb RAM / 64mb VRAM... but 9gb of HDD. Black Ops 2 is 16gb, Total War Rome 2 is 35gb. 3D models pack in more polygons and denser textures as time goes on. Indie games tend to be less greedy for space, but some of them are still biggish.
$15 not-sale price closes off a bunch of things that will likely never get lower than $20, for varying reasons. I picked up Torchlight 2 on the summer sale for 75% off / $5. Totally worth it. I've not spent more than $10 on a single game in a long time, so I understand being price sensitive... but don't shoot yourself in the foot. With your hardware as stated, you really need to start by finding games you can run, then weigh the price against the experience. Building a wishlist that only includes things you can run, then keeping an eye on sales would be a good strategy to keep your costs down.
So, more direct suggestions.
Consider spending a little on hardware upgrades... A $40 low-profile video card, even if it's in a PCI slot, will give you a lot of head room in what you can play simply for having an AMD or NVIDIA chip on it. Similarly, another $20-$40 on some RAM will open up a lot too. (do check what slots are available, what the motherboard can handle before you buy anything, and that 32-bit OSes cap out at 4gb of RAM installed / ~3.2gb showing in OS)
Go see what interesting things you can find on GOG.com. I'm not even joking. The older games they have will almost certainly run like a champ on your computer, and a lot of the newer indie games will likely be okay too. There are very few things over $15-$20, most are $10 and under, all are DRM free. (Humble Store, too...
http://calvein.github.io/humble-games/ iirc, I got this out of b12's Humble thread)
Terraria will probably keep you busy for a while. Morrowind and Fable:LC will both run for you, if you haven't already played them to death.
Play some freeware titles... hey, they're free!
Your computer will definitely Dorf, it's just a matter of how long you can run a fortress before it grinds to a halt. There's a whole continent of new-wave roguelikes out there, even more since Angband went GPL. FreeCiv, FreeOrion, MegaGlest, Battle for Wesnoth in strategy. Marathon 1+2/source, ID Tech source mods for shooters. Many many more than I can reasonably list, or recall; go check out the free game sticky.