Push away means you don't want someone's help or attention. So you're being distant. Put off means a certain behaviour or situation makes you no longer interested in being involved with someone/something. These are phrasal verbs :)

Your example for pushing me away is perfect (except a type 'yo' for 'to'.) For 'put off', the comment I would make is that 'malevolent' is a strong word. If someone were evil to the core, being 'put off' would be too light. I know he's capable, but his arrogance puts me off. I don't know much about him, but I'm put off by the fact that he tells offensive jokes. She's pretty but her whiny voice and entitled attitude puts people off. (These negative traits cause them to be disliked, perhaps outweighing other positive traits.)

The phrase "push away" simply means to push away something its a physical movement by person or thing for example. Sean pushed the door on his way out. Then we could also say the bulldozer normally push 5kgs at a time. Push away is rarely written push away, when we speak we will say pushed away. Put is to do with schedule you would usually put a event or date off.

Push away means you draw yourself away from a certain situation. And, put off means that a situation or someone makes you angry or they annoy you.

Pushing someone away is not wanting them around you. For example they don’t want to be your friend. Putting people off is ruining the mood or vibe. It can also serve as an example of someone doing something not attractive and it puts people off.