그는 떠나기 싫다는 듯이 울었어요 and 그는 떠나기 싫은 듯이 울었어요. Both are good with a slightly different connotation. As was mentioned, 싫다는 is short for 싫다고 하는, or '싫다'라고 말하는. So 싫다는 듯이 means "as if saying no". On the other hand, 싫은 듯이 just means "as if reluctant". So the difference between them is equivalent to the difference between: 1. He cried as if saying he didn't want to leave. 2. He cried as if (he was) reluctant to leave. #1 describes it vividly as if the person' expressed it himself, while #2 sounds like the speaker's observation. The -다는 듯이 form is therefore used with adjectives expressing an emotional reaction or judgment, like 좋다, 싫다, 되다, 안 되다, 맞다, 틀리다, 괜찮다, 심하다, 기쁘다, 신나다, 즐겁다, 괴롭다, and so on. For example, here are some cases where only one of the two forms is appropriate. - 그는 내 말이 맞다는 듯이 고개를 끄덕였다. NATURAL - He nodded as if saying I was right. - 그는 내 말이 맞는 듯이 고개를 끄덕였다. UNNATURAL - He nodded as if I was right - too big a gap between the action and the interpretation. - 그는 어지럽다는 듯이 휘청거렸다. BAD - He staggered as if saying he's dizzy. (we don't say something and then stagger) - 그는 어지러운 듯이 휘청거렸다. NATURAL - He staggered as if he was feeling dizzy. So -다는 듯이 is for cases where the subject says or indicates something, while -ㄴ/은/는 듯이 describes what the speaker noticed. In many cases, including your example, both are possible.

The meaning of the sentence seems clear and grammatically correct if it was written with 싫은 instead. But I think that sentence will be a bit awkward. The form of '-다는 듯이' is often used idiomatically. '-다는 듯이' is used when the subject is not 'me', it is used when trying to explain/describe the meaning of the behavior of the third person by guessing. So the sentence you posted has the following meaning: 그는 떠나기 싫다라고 말하는 듯이 울었어요.

As saying the answer first, no. It would not any changes in meaning when replacing '싫다는' to '싫은'. But there are slight differences in grammar. '~다는' and '~은(는)' in each is said '긴 동격 관형사절' and '짧은 동격 관형사절' in terminology of Korean grammar, in which depends on what noun comes after this word. This question confuses native speakers too, because this is a specialty field of Korean language. So there is no problem using either.