Here is a link to the brochures and manuals for 10D...
The camera instruction manual is 7.83MB file fourth from the bottom of the list. It's a PDF, so you'll need Acrobat Reader (free from Adobe if you don't have it) installed on your computer to view it.
There are several possible reasons for the focus problem you are seeing...
1. The electronic contacts between lens and camera might be dirty. To fix this, remove the lens and use a clean, lint free rag, dampened a little with a few drops of isopropyl (rubbing) alcohol, to carefully wipe the gold contacts on the back of the lens and the corresponding pins just inside the front of the camera. Be especially careful to avoid touching the lens optics and the mirror just inside the camera.
2. The camera battery may be old and tired and not charging fully. The BP511 and BP512 used in 10D originally are good for roughly 400 recharge cycles. Replacement batteries are cheap and plentiful, and newer ones are much higher power capacity than the originals. There are clones (non-Canon OEM) that cost very little... In the past I've bought them for as little as $8 each, including shipping.
3. The lens' focusing system may be failing. If so, it's unlikely to be practical to repair an older Sigma, so you might want to look around and consider a newer lens (I'd recommend a lightly used Canon 28-135 IS, widely available and relatively inexpensive... about $200 used... and very likely a far better lens than the Sigma you have). If you end up looking for a new lens, be aware the 10D cannot be fitted with Canon EF-S lenses. It's the last crop sensor model that's EF only, the Digital Rebel/300D that was introduced a few months later and all xxD and xxxD cameras since can use both EF and EF-S lenses. Third party "crop only" lenses (Sigma, Tamron and Tokina, primarily) all work fine on 10D.
4. The camera's focusing system may be failing. This is unlikely, but possible. Hope not, it's probably not worth repairing at this point, unless you can find someone locally to do it cheaply. 10D are a great little camera, the first DSLR to sell for under $2000 US ($1 under, originally), but now simply aren't worth a whole lot.
5. And - yes - from your description, it might nothing wrong with camera or lens.... simply be that you have the camera in AI Servo focus mode...
If that's the case, there is nothing wrong with learning to use AI Servo. In fact, I'd highly recommend it, to get the best out of your camera. I keep all my cameras in AI Servo by default, since it can be used with both stationary and moving subjects. I only switch to One Shot when I want it's somewhat higher degree of focus precision and the subject is completely stationary.
Essentially the 10D has two focus modes: One Shot for stationary subjects and AI Servo for moving subjects. You should learn when and how to use both of them. (There is also AI Focus, but this really isn't a different focus mode at all... In AI Focus the camera is supposed to decide for you whether or not the subject is moving, and then use the appropriate mode: One Shot or AI Servo. I have found this less than successful on several cameras and generally recommend simply avoiding it. Decide for yourself whether or not the subject is moving and set the appropriate mode yourself.)
You mention the "beep" and green LED, which are Focus Confirmation and only work in One Shot, with stationary subjects. The camera focuses and stops and stays at that point of focus, giving you those confirmation signals that focus has been achieved.
That's great when the subject isn't moving. But it won't work if the subject is moving. If there is movement, the locked focus will usually be wrong before you complete pressing the shutter release, and the subject will be out of focus in your shot. So you have to use AI Servo, which focuses continuously tracking the moving subject, never stops and locks, so has nothing to confirm... That's why you don't get the "beep" and green LED in AI Servo mode. Learn to use it. You'll need it if you ever shoot anything that's moving (or if you are moving while taking the shot of something stationary... such as shooting from a moving car, train or boat).
To get the best out of your camera, I highly recommend you spend a little time viewing this instruction video: http://www.youtube.com/watch?v=iAx86nblZ2g plus the other two in the series, all about the Canon auto focus systems and how to use them. Each video is approx. a half hour in length. It will be time well spent, I can assure you!