버리지다 in incorrect, 버려지다 is correct which means "to be dumped" or "to be abandoned". Now, in the song you're talking about, the 버리진 part in 날 버리진 않겠죠 actually comes from 버리다. When translated 날 버리진 않겠죠 means roughly "I'm sure you won't abandon me" but it has an uncertain tone (because the speaker doesn't know for sure). 힘겨워하다 means "someone (other than the speaker) feels tired, overwhelmed or is tough" and 힘겹다 just means "it's tiring/tough/overwhelming". 참 힘겨워했잖아요 means "it was so hard on you". Good luck.