This is a common problem especially on an opener the age yours is. That gear takes the brunt of the operation. One thing that is often ignored is the maintenance of the door itself. The springs also grow weak with time. Ten years is actually considered the normal life of an opener although most people get more.

I've had my Sears GDO for 21 years and replaced the drive gears twice. As mentioned above, maintenance on the door hardware is important to get maximum life out of the GDO. Does he door operate smoothly with the GDO, or does it shake, rattle and jerk on the way up or down?
With the GDO traveler disengaged, make sure the door operates smoothly and is in balance with the springs. With the bottom of the door at about chest height, it should not have to be held to remain in that position.

I had the same problem with my drive gears, and I've replaced them. But now the door won't open more than 3 inches, so here's my question: if my door wouldn't remain in waist-high position when the traveler was disengaged, am I having an issue with my springs? If so, can I repair that on my own, will I need help, or should I just replace the whole system?

I'm a firm believer in leaving the springs to the professionals. They can be dangerous. I think you should call out a door technician and let him recommend what you need. Talk to some neighbors and see if you can find someone they have had experience with.

It's odd - once the new gear was in place, I tested the opener without the exterior casing or the sprocket cap, and got no results. I put the casing and cap back on, and everything works now. Either I underestimated the importance of the casing and cap, needed to give my opener some time to resync with the power on, or just caught a break. Thanks for your reply about the springs - I tend to agree.