I apologize for my late reply (unavoidable circumstances). I found out that I didn't have the right screwdriver size. I just needed something that would give me enough torque.
I am having another problem, though. I wish I had a digital camera to take pictures and show you guys, but I'll do my best to explain it.
The SFF Acer happens to have used a heatsink that screws straight into the motherboard, whereas my new motherboard has one of those plastic brackets that is screwed into the motherboard, and it just requires a heatsink with those clips that the SFF heatsink don't have.
Since my motherboard (Biostar A770E3) is socket AM3, I assume the AM3 heatsinks with the metal clips should be the correct width...
Also, can I get a heatsink with a 3 pin connector when the connection on my motherboard is 4 pin?