Septic tanks typically do not have a "breather". They generally have one or more manhole access points at the top. My tank has one manhole in the middle. You can sometimes run a snake from the house through the (typically) 4 inch sewage pipe that flows into the tank.
As for the reason your tank is overflowing, could be a blockage in the outlet tee, could be a bad distribution box, or could be a clogged field. You can always get the tank pumped out, this makes it possible to do some diagnostics on the tank, or you can dig up the distribution box to see if it is plugged or backed up, which often indicates a failed drain field.