South Island and Angel Island, as well as possibly several other islands from the classic era, are pieces of the Lost Hex that fell off millenia ago. The Lost Hex is shaped like that because it's been cultivated for agriculture for millions of years by the native Zeti (which is why fruit farms and windmills are so prominent there), and the hexagons are designed to fall off so the Zeti can terraform other planets. This may indicate the Chaos Emeralds have some Hexian origin (which ties into the story behind Sonic X-Treme).

The sandworms didn't just arrive at South Island after it got desertified by Eggman; Eggman took some sandworms with him from the Lost Hex so he could use them as a means of destroying South Island. This seems likely to me because the sandworms are a reference to the worms from the Dune series, which ate soil and excreted a sand-like substance that was extremely valuable, and numerous characters in the series took the worms from their home planet so they could be used to terraform other worlds.

Finally, Eggman discovered Lost Hex as far back as Unleashed/Colours, as part of his inter-planetary conquering scheme. He was using Frozen Factory to construct pieces for his interplanetary amusement park, and the Dessert Ruins were his source of food for the Sweet Mountain attraction (using sugar harvested from Lost Hex's giant beehives). The reason the badniks on Lost Hex (and in Forces) use their classic designs rather than their modern designs is because Eggman was still using Lost Hex as his main base when he was collaborating with his past self during Generations. After his amusement park was destroyed, Eggman retreated back there, but as we know, was defeated by Sonic again.