They often force air down pipes which can cause the plumbing in your home to back up.
That said, it sounds like the problem could just be the disposal. Your disposal should have come with a small wrench that you can insert in the bottom to turn the blades inside. If the disposal is getting power but not moving the problem is probably there.
Did a glass get broken and fall in? That often caused problems.
the pipes aren't clogged or leaking.
I bought a new one for about $70 and replaced it. I'm a hack at such stuff, so it took a couple of hours. A good handyman/woman can probably do it relatively inexpensively.
In a previous house, my disposal would turn on, but it wouldn't clear anything. I unhooked the pipes and cleared out a gross clog with a coat hanger. Worked perfectly until the next clog.
To clear it, and it may need to be reset. I know if I havent run the disposal, the draining of my dishwasher can cause the water to back up into the sink (doesnt get to the point of overflowing onto the floor, tho).
Heres a YouTube showing the crank and reset, and clearing clogs:
If none of that helps, then I suspect theres a problem further down the line, or maybe the vent stack thing? I hope that it doesnt come to that.
They jam, clog, and take up valuable space under the kitchen sink. Personally, Id disconnect it and call it a day.
That sounds like a good idea, not sure my skills could cope however.
He took the disposal out and cleaned it out.
BTW - replacing a disposal isnt so easy. First, you need to make sure the replacement you get matches the configuration under the sink. Best if you can replace with the existing model.