[wa/id] Do I Have To Pay A Cleaning Fee If I Can Prove The Landlord Never Cleaned The Carpet Before I Moved In?

For clarity, the property is in Washington but the rental agency is in Idaho.

The property rental company has done a lot of shady stuff already, including blatantly violating WA heating code (tried to rent me an apartment that had essentially no installed heating until I sent them the code and they had to install heaters to avoid getting reported to the housing authority). They've lied to my face that they repainted the apartment but I have photos and videos from when I cleaned the walls and ceilings that show build up of tar and grease. The most recent incident is I have evidence they didn't clean the carpets before I moved in despite telling me they had. This includes a candle wax stain that is fairly large and a strong odor.

It is stated in my lease that I cannot wash the carpets but I'd asked several times for the wax stain to be dealt with at the very least but was ignored. I washed the carpets and found that there was no way they'd cleaned the carpets and that there was likely mold growing in it. In my lease it also states that I'll have to pay a $120 carpet cleaning fee upon move out (I have only one small room with carpet mind you in a one bedroom apartment) and I want to know if I can dispute this charge when I move out by showing they did not clean the carpets prior to my move in. That or at least that they're overcharging for cleaning. I know that the fact that I technically violated the lease by washing the carpets complicates things, I have no idea though if it is effected by the fact that they were hiding evidence of lying and thus cancels out. Any idea on how to handle this and where I stand would be appreciated.

Note: I know I should have backed out rather than moved in but the place I was living before was dangerous and on the brink of being condemned and it was move to this apartment or become homeless. I'd rather deal with a slum lord than the streets at the end of the day.