I have almost no accent (or the generic American accent) and most people don't know I am from the South unless I tell them. I have lived all of the US and most of my time in Texas as an adult is the DFW area which is so over run with Yankees that an actual Texas accent stands out. When I was kid a had strong Texas accent but I grew out of it by the time I joined the Marines.

I do you the southern speech pattern though and say thing like I'm Fixin To, y'all, around about and catawampus and I pronounce wrestling wrastling. DO you still have your accent?

As for the rest I simply don't care enough to debate the issue but as someone who speaks to people from India on a regular basis I can tell that fluent is a stretch for many of them and if the ones who are fluent use phrasing that I would not so much British as odd.