Broken how? As long as the drain is still connected to the tank the water will go into it. Where it goes after that depends on where and how it is broken.
The "septic system vent" you are referring to is not actually attached to the "tank" (that would be the buried, holding chamber in your yard) at all. It is attached to the plumbing in the residence...
If it is full, you can't put more in it, full is full. Flush the toilet, it is going to come up in the shower.
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There is a new discovery called "WATER SEEKS IT'S OWN LEVEL" . If the septic tank...
Either the main line is plugged or collapsed or more likely the septic tank needs to be pumped out. The tank needs to be pumped out every few years. This varies according to size of the tank and how...
If the old tank is still in place and usable, probably not. If it has been removed then yes. Regulation may have changed and in some places septic tanks may no longer be allowed.