Will Apple repair my phone if they've sent me a faulty one?

Last year I had Apple repair my iPhone 5's sleep/wake button. They sent me back a different phone, I could tell because the serial number was different. I had no problems with this phone until a few weeks later. At first these squares showed up on my screen, this wasn't a huge deal but it progressed with my screen going a bit dark and being unresponsive to touch. I managed to just use Siri during these periods but then it stuck like that. Now my screen just goes wild. If I can manage to get it unlocked, it'll switch pages by itself, open apps, and just is generally impossible to use. I can't even use Siri anymore because the screen will cancel out my mic. I've tried restoring it from a backup, updating the OS, factory restoring it, and anything else I could think of. I put the phone aside and used a family member's old android. I contacted Apple live chat sometime in November asking if they'd repair my iPhone and they asked me to try restoring it from a backup, another time. I tried. Didn't work. I had a lot of things on my plate at the time and I put it off until recently. Now I find out my Apple Care ran out sometime between December and now. I've been trying to contact Apple Support to no avail, as it is seemingly impossible unless you have Apple Care.

Does anybody think Apple would repair my phone, and how the bloody **** am I supposed to contact them?

Thanks in advance, I'd really appreciate the help.