This is crazy, but I found out what was wrong with my lens! I was at a camera show today asking the people at Sigma and they didn't know what was wrong, but showed me how the lenses with Nikon mounts have a mechanical aperture control that operates on a spring and is defaulted in a closed position. I am new to using lenses like this, so wasn't aware of it (I know, crazy).
Anyways, at home I checked the speed booster. Moving the aperture control dial with no lens on, I could see the speed booster had a similar thing that moves back and forth. Obviously, it has to be in the right spot to hit the one on the lens. I tested my lens, it still didn't work. Then I marked how far I had the lens screwed on (matching the words). Taking the lens off, I held it just in front of the speed booster in the same position, and noticed the two little aperture parts that should push each other were about a centimetre (0.39") off.
Turns out, I wasn't screwing the lens on anywhere near tight enough to the speed booster. I screwed it on again, pushing it hard (and feeling like it was forcing) the extra distance. Low and behold, the lens now works. And I feel really silly for not realizing this before (especially now realizing that when screwed on correctly, two obvious white dots line up). I think I was just afraid of breaking the lens or speed booster by screwing it on too tight, since it had already felt like it had 'clicked' into place, when clearly it hadn't.
Don't know if this will solve anyone else's issue, but it was strangely simple, and you won't believe how relieving for me to finally get this to work.
Thanks again for everyone's help!