iphone 5 s is randomly opening applications or typing characters in text messages and imessages

I recently had an iPhone 5s replaced (in September) through an insurance claim after damaging my first phone. With this phone and recent software updates, I have been having periodic issues, that have now gotten extremely bad. My phone randomly launches applications and also randomly types characters in text messages and imessages when in use. The other day, for example, I had just sent a message so the phone was not in a locked state, and it decided to randomly facetime one of my contacts. Any thoughts on what might be happening or what to try? I powered the phone off completely, and also tried re-setting my phone to default settings. But I am also not sure if I did this correctly. It did not delete any applications or anything, just put my settings back to original state.

Thanks