These pairs are called doublets.
The meanings are in fact related. Sometimes the relation is obvious, as in arrêt "a stop" and arrête 3cs pres. "to stop". Sometimes it's buried, as in magasin, originally "storehouse", and magazine ultimately from the same root — via either figurative "storehouse of information" or because the periodicals were originally inventories of magazines. A similar history can be spelled out for droit / droite (and conversely sinistre / senestre).
Some of them are what you call "convergent evolution". I'd prefer "parallel evolution" since "convergent" suggests separate origins, whereas doublets share an origin, but the path the word has taken is different — like twins separated at birth who find each other in middle age.
Sometimes the path is different because, as you suggest, a local dialect influenced one form. Sometimes it was borrowed twice from different languages, as in magasin / magazine (both from Arabic, but the former was taken via Italian and the latter via English).
Sometimes the divergence is due to different grammatical forms in the language of origin, i.e. two derivations of a root in Latin gave rise to two different products in French.
And sometimes they're simply different forms according to the modern rules of the language in question — although in this case they aren't called "doublets". This is true for arrêt / arrête (and other conjugations of arrêter), which we call "inflections" or "derivations" as the case may be.
You should be able to find lists of examples now that you know the technical name. The bulk of them could be considered "finite" in the sense that fossil fuels are finite: more can be made, but it takes a long time. However, even today words are always splitting off among different groups of speakers. But the standard form of a language as codified in a dictionary is unlikely to recognize two variants of a new word when it's slow even to recognize one.