I don't know what you mean by "not a chipped model".... If it's an AF lens, has electronic contacts, and since it's an HSM lens it most certainly is... Then it's got a controller chip in it to operate the AF and the aperture. As such, there is no reason you shouldn't be able to stop it down on your camera.
You did the right thing cleaning the contacts.... usually that clears up lens to camera communication problems.
No, don't use a pencil eraser on lens-to-camera electronic contacts. The most recent white paper from Canon about clearing error codes specifically recommends against it, though I know their techs have suggested it in the past. Now they are saying to not use an eraser because it risks damaging the gold plating on the contacts. However, the reasons I think it's a bad idea to use pencil erasers are that often the problem with contacts involve oils... and pencil erasers are made from vegetable oil, so will do little or nothing to help remove any oily coating on the surfaces, might even make matters worse. Also, erasers shed particles that are risky because they might get into camera mechanisms.
The reason to use an eraser carefully on some other types of electronic contacts is to remove corrosion. That's fine on chrome plated, stainless steel, etc., contacts that can become corroded. However gold plated contacts cannot corrode (gold is entirely non-corrosive, which is precisely why they go to the trouble and expense to use it on the lens-to-camera contacts).
If tightening the screws doesn't work, I'd be contacting Sigma about sending it in and getting it fixed.