(I could make some jokes here, but I do not want to annoy the locals) Sorry if I ask, but have you actually ever been to Germany or is your knowledge limited to the TV news?

The opposite is true, the closest to English would be the low German dialects that are similar to Dutch. As a matter of fact, I guess only Austian German might be more far away from English than Swiss German.

Sorry, based on what? Give some examples. I do not see too much difference in the grammar. "Educated Swiss German speakers" use grammar just as complex as High German. If you refer to some simplified expressions you hear on the street: not all Germans in Germany use all forms in every day talk either...

The OP is not aware of the differences, so you should not mislead him: Swiss German is never written with the exception of txt-speech and graffity. High German is the official language here, all documents are written in it.

So from any angle, making a basic German course is the start. The local dialect will automatically follow when you live here. You will figure out that the locals use different terms for "sandwich" as your German book without problem.
If I would move to the US, I would also simply brush up my English and not look for some course on "US specific terms".