In this case, I think '이사하느라 바빳어요' is much more appropriate. '이사하기 때문에 너무 바빴어요.' feels too formal and long. I think '~하느라' is usually used when hard working(especially in this case, 이사) or doing sth too much results in negative situation. ex) 일하느라 아내의 생일도 챙기지 못햇다. 게임하느라 밥도 못먹었다. (doing too much) A: 너 왜 어제 수업 못왔어? B: 밀린 과제하느라 못갔어.

-기 때문에 is mainly used for formal writing. It rarely used in colloquially. It is replaced by -어서, 길래, 니까 in colloquially. For negative clause, 느라고 or 는 바람에 is more natural. ~느라고: The first clause is a reason or cause for the second (usually negative in nature) Equivalent to "Because of that..." Usage: -Can be shortened to ~느라 -Action in the first clause continued partially or completely with that of second -Don't: use positive clauses afterward - it sounds unnatural -Don't: use propositive/imperative forms -Don't: use past tense ~았/었 -Use idiomatically with 수고하다, 고생하다 -ONLY use verbs that show the (1) will of the subject, (2) force of energy, (3) time -Subject stays the same in both clauses ~는 바람에: The first clause (usually a negative influence on the second) describes the reason or cause for the second clause Equivalent to "Unfortunately, as a result of..." Usage: -ONLY used with verbs -Second clause is ALWAYS past tense (because it's describing something that has happened) -Don't: use positive clauses - it sounds unnatural - this expression is for negative things that have happened (but very rarely something positive can be said - only if it was completely unexpected or unintended) -Don't: use propositive/imperative forms -Don't: use past tense ~았/었 -Difference: ~아/어서 = a simple reason, ~는 바람에 = an unintended result e.g. 어제 늦게 잤는 바람에 아침에 늦게 일어났다 (X) 어제 늦게 자는 바람에 아침에 늦게 일어났다. (O) = I slept late yesterday so I woke up late in the morning. 무릎을 다치는 바람에 병원에 갈거에요. (X) 무릎을 다치는 바람에 병원에 가게 됐어요. (O) = I hurt my knee so (because of that) I ended up going to the hospital 야구를 보는 바람에 숙제를 못했어요. (X) This sentence is wrong because watching baseball was not an unexpected occurrence or something I wouldn’t have hoped for. So since the action is our own, we have to use -느라고. 야구를 보느라고 숙제를 못했어요. (O) = I couldn’t do my homework because I watched baseball. 컴퓨터가 고장이 나는 바람에 숙제를 못했어요. (O) = My computer broke so I couldn’t do my homework.

The main difference is that -느라(고) indicates the speaker's direct involvement with something, while -기 때문에 can be about any kind of a reason. -느라고 typically gives some work or even one was tied up with which caused something, like preventing some other important thing, rather than a logical reason. Also, with -느라고, the cause and effect should be closely connected in time. You can't say 오늘 낮에 일을 너무 많이 하느라고 저녁에 피곤할 것 같다 (mismatch of time) for example, whereas 오늘 일을 많이 했기 때문에 [많이 해서] 저녁에 피곤할 것 같다 is fine. In your example, only the first one sounds natural. - 이사하느라 [or 이사하느라고] 너무 바빴어요 - natural, common way of saying it. - 이사하기 때문에 너무 바빴어요 - neither 이사하기(present) nor 이사했기(past) works as well as 이사하느라. Here are other examples for which only one of them is natural. - 내일 출장 가기 때문에 지금 좀 바빠 => Can't say 출장가느라고 because 출장 is a future thing yet to come. - 겨울엔 날씨가 춥고 건조하기 때문에 감기 환자가 늘어난다 => -느라고 can't be used for non-personal reasons. - 무거운 가구를 옮기느라고 고생했다 => 옮기다(moving it) and 고생하다(have a hard time) are inseparably connected. -느라고 works well for such cases.