Why are there not more foreign languages and influence felt in Westeros? Considering how close the Free Cities are wouldn’t it be logical?

Eh, it’s a premodern society. 90% plus of the population probably never travels more than 10 miles from their birthplace, levels of international commerce and travel are much, much lower than in later epochs, levels of literacy are low and there’s no printing press, etc. 

There’s influence, but it’s contained to the port cities and among the nobility.