The Air Admittance Valves do what they are called=======Admit AIR into drain piping.
A faulty one would NOT have 'signs of overflowing'
They are installed 'downstream' of sink/shower 'P' traps to prevent 'P' traps from getting sucked dry when drain waste tanks. They OPEN when draining and then are plugged/pushed closed (depending on design)
The cheap ones (like in pic) use a tang on flapper to pull them closed (better models use a spring for longer life/more positive closed)
Even bathroom sinks that drain to black tank will have one
Black or grey waste tanks can stink----grey sometimes worst due to lack of back flushing systems like used on black tanks
So if toilet bowl HOLDS water and toilet floor flange seals off and sink/shower 'P' traps have water THEN Air Admittance Valves are suspect.
Tape a bag over them and then go about dumping running fans etc and see of you still get a smell
NO smell replace ALL AAVs
Still smells--------then will need to smell down around toilet floor flange and check that waste tank vents have not slipped down below roof line and are allowing tank venting to be pulled inside roof (hence inside rv)
That means up on roof and pull vent caps to visually see where end of vent is in regards to roof lone