I think Roman Carthage was basically a continuation of Punic Carthage. Punic continued to be spoken there (and actually continued to spread in the region into early Imperial times, with Punic poetry being written into the late 4th century) and the city was among the most important in the Roman Republic and later Roman Empire. Only after the Arab conquest was it sidelined. Seems inevitable in a no-*Maghreb Islam TL that Carthage retains its status (thanks to economics and its religious supremacy) and maybe even in a post-Islamic TL Carthage might replace Tunis and retain its importance. My ideal post-Roman polity there would span at least from the Gulf of Arzew to the border of Cyrenaica and incorporate Sicily (ideally the whole Maghreb, Sicily, Andalusia, the Baleares, Sardinia, and Corsica under this polity plus the Saharan portions of Niger and Mali), with its national language the African Romance dialect of Carthage. In the modern age, Carthage would become like Rome, Madrid, or a more developed version of OTL Tunis, Algiers, or Alexandria.
As to how to save lost cities, a lot of Mississippian centers, most notably Cahokia, would have benefitted from better soil practices and a better source of protein. Beans (which fix nitrogen and have protein) were not as widely grown as they could've been, while no source of animal manure was available. Domesticated ducks and turkeys in the Mississippian civilisation would've been huge since they give both protein and are a good source of manure. So many centers might've survived in the Mississippi Basin. Drought and European disease would've nailed them in the early 16th century but there would've been no Empty Quarter and De Soto would've encountered much more powerful states (and possibly never returned considering Spanish actions). Much of the Eastern United States if colonised would've been more like Latin America in terms of demographics.
