i think it would be helpful to you how they are translated in english.
'~(한)후(에)' simply means 'after',on the other hand,
~(한) 다음(에) can be translated as 'the next' or 'the following'
the usage of them will be a bit different depending on contexts, but
they all are the same meaning at least in your sentences.
even if you put '에' right after '후' or '다음', the meaning will not be changed, also
makes it more natural.
i think '다음에' is more natural and common in spoken,
and you should get used to using these patterns, ~(한) 다음에/~(한) 후에
so your second sentence should be '그녀는 출산한 다음에 힘이 다 빠졌습니다'
not '그녀는 출산 다음 힘이 다 빠졌습니다'
here are some examples of their usages.
점심 식사 후에 난 낮잠을 잤다.('after')
점심 식사 다음에 난 낮잠을 잤다('the next', it's very unnatural)
shoud be ' 점심 식사를 한 다음에 난 낮잠을 잤다'
(after lunch, i took a nap.)
30분 후에 전화 드리겠습니다.
30분 다음에 전화 드리겠습니다.(also unnatural)
(i will give you a call in 30 minutes)
다음에 만나요
후에 만나요(we can understand this, but quite unnatural)
(see you later)
다음에 고치겠습니다.
후에 고치겠습니다.(unnatural)
(next time, i will get it right)
다음에 차나 한잔해요
후에 차나 한잔해요(unnatural, could be a dialect)
let's have a tea together sometime.
Also '다음' can follow nouns, but '후' doesn't.
다음주 next week
다음달 next month
다음 정거장 next stop