Thinking of the literal translation helps.
整 is 'entire, whole'. 个 is a widely applicable singular measure word, hence 'unit'. So 整个 is, literally, 'the whole unit (of), the entirety (of)', as opposed to 'a proportion (of)'.
我整个身体都不舒服。My whole body feels sick.
However, I don't think 'whether the object is having parts or not' matters for applying 整个 (on the contrary, it matters for 全). In the following examples, 'the ground' and 'the afternoon' are not 'obviously decompose-able':
白雪覆盖了整个大地。The entire ground is covered by white snow.
他一整个下午都在看书。He was reading books for the entire afternoon.
全 means 'all, every'. It is applied to a collection, or a unit that has 'obviously discrete parts'.
In this sense, as the other answers already mentioned,
整个身体 means 'the entire body' and
全身 means 'all parts of the body'.
我全身都不舒服。All parts of my body feel sick.
(As for why
全身 is used instead of
全身体, it's probably because 全, 身 and 体 are all bound morphemes, and people tend to construct words with the least number of bound morphemes necessary.)
As you may notice in this example, although 全 is applied to a collection, the word after 全 is not always a 'group noun'. Actually in most cases, it is only in the semantic but not the syntax/word formation. For example:
全球金融危机 The global financial crisis
他全天都在工作。He is working all day long.
In these phrases, the knowledge (in the extent of Chinese language, not philosophy) that 'the globe' and 'the day' can be further broken down is implicit. On the other hand 'The afternoon' is considered atomic, so you can only say
整个下午 but not