:/ that sounds really uncomfortable.
You might have more success if, in addition to the thermostat, you can point out WHERE you have found air coming in. Sometimes its easy and you can see daylight (and thus air can come in). I've definitely wandered around with damp hands trying to work out where that air is coming from.
I was renting from a friend in a tiny, adorable, uninsulated poorly maintained house with air leaks EVERYWHERE. The window plastic made a big difference. I put it on every window I could. Afterward, I could watch the plastic move a bit in response to the wind, but I didn't feel the breeze anymore! It was amazing!
I successfully installed new sweeps for regular doors. Youtube might be able to help with suggestions for weatherstripping sliding doors? Some supplies aren't THAT expensive and it might be worth it if the landlord isn't helpful.
There is also a really cool thing called Moretite rope caulk, which is basically like a GIANT wide strip of poster tack that you can make any width or length you need to. The first time I tried to use it, I was trying to stuff it into cracks and airgaps. The strategy that ended up working was to actually just leave it in big chunks and completely cover over the gaps. I'm using it to cover gaps where some casement windows don't close completely because the frame is warped a bit. It's removable and doesn't leave any residue.