Should Dorne and the North have a slightly different alphabet/linguistic system compared to the rest of Westeros due to Rhyonish/First Men influence?
If we’re going to start with the idea that all of Westeros speaks the Common Tongue as their primary language as GRRM does, then there’s a limit to how much variation there should be.
A different alphabet? That really depends. The Old Tongue of the North had a runic script but was primarily an oral tradition, so in a scenario in which the Old Tongue survives, I’d imagine it would be transliterated into Andal script. Given that the Rhoynar were a more technologically advanced and literate culture than the First Men, I would expect a Rhoynish script to survive if the Martells hadn’t banned the Rhoynish language.
A different linguistic system? Well, given that the Andals had been in Dorne for thousands of years when the Rhoynar arrived and then there was an intense period of intermarriage and cultural borrowing, I would expect the emergence of a creole language that combined lexicons and grammars from the two language, with perhaps the elite learning “proper” Andal or Rhoynar or both (depending on the House’s heritage or their desire for political advancement at Sunspear). Given that the North avoided Andal invasion altogether, I would expect bilingualism to be the more likely scenario, with Andal becoming the language of trade and diplomacy, spoken by merchants, sailors, the nobility, and more common in the White Harbor area due to the transplantation of the Manderlys, but as a secondary language, with the Old Tongue as the primary language and the only language of the vast majority of the population in the interior.