I've had rescue cats for the past 40 years - hence the name madcatwoman - and one thing is for sure, every cat is different and there's no 'One size fits all.' I can only offer a few tips:
Cat pee smells like bleach and vice versa. First off, I would buy two new litter trays. Clean them both with bleach and rinse mega-thoroughly. Your cat will smell the faintest traces of bleach - they have fab noses - and hopefully, should put 2 + 2 together. Et, voila! Or not!
I use Tigerino which is clumping litter, so you just need to lift the lumps and all is good. No horrid, smelly trays. (Cheapest on Zooplus - 10kg lasts a month or more.) That's when it is cleaned out and trays bleached etc. twice a month.
Cats are weird - what suited in the last house, may not suit in the new house. Which is why I suggest new trays and different litter.
Good suggestion regarding trays next to outside door - maybe Pusskins wants to try a new toilet vibe ... like going outside.
Please give it time. These things can take a good while to get going. Cats are, in some ways, resistant to change (but you've just said get new trays!) I know, I know ... but you're dealing with a cat ... and only a cat understands The Way of the Cat.
My heart goes out to you, but be patient. I'm sure when she/he settles down they will repay you a thousandfold. All of mine have (little blighters) and I couldn't love them more.
Best of luck.