There are two "contraptions" inside the tank of your toilet. One for filling the tank, one for flushing it. Your symptoms sound as though the fill valve is not shutting off as it should and water is running through the overflow into the tank causing it to flush.
There are several types of fill valves, but most work by having a float rise on the water level until it shuts off the flow. Try adjusting your float so that it puts the fill valve in the off position at a lower level.

FWIW.....I'm responsible for a total of six toilets at two residences plus a few elderly neighbors. Over the years, I've replaced all the original mechanisms in each of them with Fluidmaster. They've all worked perfectly the first time, haven't broken, are easy and quick to install, and are available cheaply almost everywhere. Parts, too.

I don't know about your particular toilet/tank arrangement, but if one of these units would fit it, I'll bet you're about 15 minutes away from solving your problem.

I'd actually go the other route, just my two cents, if the fill valve keeps running, toilet is going to over flow, if the flap isn't working properly, it will let water seep by, causing the fill valve to operate to fill tank up again, the noise you are hearing is probaly the tank filling up with water.

Sounds like the flapper isn't sealing. I just replaced with a Fluidmaster "Bulls Eye" - nice fit. If the old flapper was deteriorated, then might want to rub your thumb around the flapper seat to remove any remaining rubber gunk.

Alternative: If the water level in a full tank is right at the top edge of the down tube (center vertical tube), then adjust the float so it shuts off before water gets right to the edge. With fluidmaster, usually do this by either squeezing a metal clip (below the float arm lever) and moving clip downward a bit, or alternatively (other models) by using a screwdriver and do a couple turns.

Thank you! It was the flapper we changed out (twice). I'll try the other suggestions and let you know. Thanks again!