Americans actually have British accents

I was told by a teacher of mine that in the really rural south, where people didn't really leave or come to the area, that the accent that they have is closer to the accent that Shakespeare probably spoke in.

Yeah that's what I've heard too. The original british accent is the American southern accent

Country? Because that's all I hear down here

It is unlikely that the british accent sounded anything like modern day american. However it make have been more rhotic in nature (prolonged R sounds) as britain had been invaded by the french (and the rest of europe at various points) who have a more rhotic way of speaking. Scottish accents are also considered a rhotic accent, as are many british accents that currently exist (such as geordie accents). Based on these observations I would hypothesise that british accents have never sounded like any modern american accent, although we (british) may have lost the prolonged, rolling "R" sounds associated with american accents, while americans simply retained it and emphasized it. Also the influx of africans in early american culture would have encouraged the development of the rhotic sounds due to the difference in tongue muslces and structure, and modern American is just as likely to have stemmed from that influence as from the native accents carried over from britain and ireland.