First & FAST. Fill up a soda bottle with water.(basically a cup of water) and pour it down every sink drain and FLOOR DRAIN.( Usually that is located near the water heater tank or between the laundry room and the bathroom or in the kitchen (on the floor) as a 4" chrome disc full of round holes or a 4" circular black plastic grate...it stops big stuff from going through but allows water to go down. Many times a piece of furniture or rug covers it over. It has what is called a Pee trap, it stops the smell of Pee or other sewer gases to blow up into the room but still allow water to drain down and out of the house. If you look under the bathroom sink or the kitchen sink you will see a "U" in the plumbing. Water sits in the "U" that keeps the smell in the sewer and allows water to drain down.
. So, how many sinks do you have? Remember the shower you now use as a closet instead. If there was suppose to be water used in it then it has a Pee trap.
This is the official Plumbing term. Every drain has a Pee trap. Any can dry up. Adding a cup of water will hold it for a Year or longer. You can never over fill them so add a cup or 100 gallons makes no difference. A cup is all you need.
You see, everytime you flush the toilet that "chunk of water" enters the septic or sewer system and that displaces air that is stinky and forces it up the dry P traps like a "POOF" of stench. Does not smell like Pooh or Pee because there is shampoo, hair conditioner, bleach, cooked turnips, etc is poured into the drain and it is all decomposing. That is the foul odor you smell. You FIRST GOT TO STOP IT FROM ENTERING THE HOUSE. Somewhere, some drain has not been used ever. Usually it is a floor drain as there was a time when all floors were wet mopped. Now with dry mops and vacuum cleaners and brooms it is forgotten so they dry up.
Then air out the house.
I worry less about the toilet malfunctioning . It just has a small pee trap and not a big pool of water to defecate into. It is a style of some toilets.
If the smell is coming out of the bowl, either get a different style. Maybe the landlord has a different toilet sitting around and you can mention the fact that the stink is coming out the bowl. He knows that is not the way it is suppose to work. (or add a 1 quart pot of water to the bowl after you finished flushing so there is more water in the bowl.) You have to slowly add it otherwise it will "flush". I have never seen a toilet that did not have its own pee trap because that is essentially what the bowl is.
Airing out the house means washing everything that is cloth. As carpets and drapes and sofa's and ceiling and wall paint(yeah, wall paint holds onto stink like cigarette smoke or sewer smells) and other stuff. It sticks like honey.
You will need to have the windows open and air out the house....and fight off the flies...to them this smells like a meal. And have bowls of vinegar sitting on shelves or ledges and allow the vinegar to evaporate and take the odor with it as it evaporates. Use white vinegar...otherwise the other vinegar's(wine, apple cider,cherry) attracts fruit flies.