Learn English – Gerund (being sick) at the end of the sentence

SHORT ANSWER:
Neither version expresses two independent thoughts: the participle clause modifies the meaning of the main clause.

LONG ANSWER:
An independent clause requires a finite (tensed) verb. A participle^† clause/phrase^‡ like this cannot be an independent clause; it must be parsed as subordinate, either an adjunct (modifier) on one of the constituents or a ‘supplement’ to the main clause. The term ‘supplement’ is that of the Cambridge Grammar of the English Language, which gives examples:

i  a. His hands gripping the door, he let out a volley of curses.
  b. This done, she walked off without another word.
ii  a. Realising he no longer had the premier's support, Ed submitted his resignation.
  b. Born in Aberdeen, Sue had never been further south than Edinburgh.

The [italicized] non-finites are supplements with the main clause as anchor. Those in [i] contain a subject, and belong to what is known as the absolute construction, one which is subordinate in form but with no syntactic link to the main clause. Those in [ii] have no subject, and are syntactically related to the main clause in that the missing subject is controlled by the subject of the main clause: it was Ed who realised he no longer had the premier's support, and Sue who was born in Aberdeen. In neither [i] nor [ii] is there any explicit indication of the semantic relation between the supplement and the anchor. This has to be inferred from the content of the clauses and/or the context. The natural interpretation of the supplement in [ib], for example, is temporal (“when this was done”), and of that in [iia] causal (“”because he realised ... ”). Both constructions allow gerund-participials or past-participials. [1265-66; boldface emphasis mine]

The position of the supplement is significant only if the interpretation is temporal: the supplement and the predicate of the main clause have to be set in their narrative order. But a causal supplement may be set in any position.

There is an additional twist when the gerund-participle is being. The ordinary way of expressing the fact that John was sick when he went to work is to employ the bare adjective, as TRomano suggests. Adding the otherwise redundant participle implies that you have some more complex meaning in mind, and consequently it virtually compels a causal interpretation:

Being sick, John went to work = Because he was sick, John went to work.
John went to work, being sick = John went to work because he was sick.

I don't think that is what you intend!

The version with getting is naturally interpreted as a temporal: John went to work and on the way started to get sick.

^† _{When the -ing form is employed this way it is called a present participle. The term gerund is reserved for situations in which the -ing form is employed as a noun or as the head of a phrase which acts as a noun. If you wish to speak of the form rather than the use you could follow CGEL and call it a gerund-participle, or you could call it simply the -ing form.}

^‡ _{Traditional grammar treats constructions like being sick as phrases because they have no overt subject; some modern grammars treat them as clauses because the subject is inferrable from context. It doesn’t really matter what you call them, as long as you and your readers have the same understanding of the terms.}