도저히 and 절대로 - they are not the same and almost never interchangeable.
도저히 describes when something won't go as one wants despite a great effort or wish, like "no matter how I try".
절대로 shows one's will and determination not to accept or allow something, like "by no means" or "never".
도저히 is exclusively for negative sentences. 절대로 may be used in positive context too, though negative is the norm.
- 이 창문을 도저히 열 수가 없다: I can't seem to open this window no matter how I try.
- 마지막 문제는 도저히 풀 수가 없었다: I just couldn't solve the last question for the life of me".
- 아니야, 절대로 그럴 리가 없어: No, that just can't be true!
- 내가 살아있는 한 절대로 그렇게는 못해: I will never allow it so long as I am alive.
- 이 경기는 절대로 이겨야만 한다: We absolutely must win this game. (positive context)