In 我只是为了让你觉得我在关心你，然后以此在逃避, who is escaping? Does it refer to an escape route?
You are almost there.


  我只是为了让你觉得我在关心你，然后以此在逃避


I just want you to think I care about you, 
and use this(refer the act"you think I care about you" or "I care about you", both could be right. Can not know since the information is not given) 

to escape(from something. [Escape from what? it's not given]).