1)
어쩜 is short form of "어쩌면". It means "perhaps" or "how" according to the context.
In this sentence, 어쩜 means "how" because it is an exclamatory due to "이렇게 ~지".
2)
딴 얘길 시작할까
딴 = 다른 means "different"
딴 얘길 시작할까 = 다른 이야기를 시작할까 means "Shall I start a different story?"
3)
눈물이 차올라서
차올라서 is composed of
차오르(다)+아서
=> 차오르아서 => 차올아서(ㅡdrop) => 차올라서(ㄹdouble)
차오르다 is a 르 irregular verb
차오르다 means "fill to some level"
아서 is a connecting particle used for describing cause of an action
눈물이 차올라서 means "because tears are filled up"
*르 irregular verb
Verbs and adjectives with the last syllable 르 have a special form when conjugated with all A/V + 아 / 어 / 여 patterns.
The choice of 아/어 follows the syllable before the final 르. The ㄹ is doubled and the ㅡ is dropped. E.g. 부르다 becomes 불러, and 다르다 becomes 달라.
4)
울면서 할 줄은 나 몰랐던 말 = 나 울면서 할 줄은 몰랐던 말
it means "Words I didn't expect to say while crying"
"Words I used to not know, I said while crying"
means
내가 몰랐던 말을 울면서 했지.
cf.
~ㄹ/을 줄 모르다 means "does not know how to do"
저는 스키를 탈 줄 몰라요 = I don't know how to ski
저는 그것을 할 줄 몰라요 = I don't know how to do that