What is done and whether 10w-30 is a good idea depends on where you are. In Canada,no. In Mexico maybe. With the 100 days of summer arriving in Florida, it would be fine here.
Today's computers have a lot of inputs so the first issue is whether any error codes have been set.
Next I'd observe the throttle positions recorded vs actual reading (your GC is "fly by wire", there is no physical connection between foot and throttle body).
Both can be done using the OBD-II connector and proper software/dongle (from Amazon
for under $25 if you have a notebook/netbook/ultrabook PC - plug).
Without that you migght as well take back to the dealer and ask for help, there is not much you can do.