#6 THWN copper, with green insulation would be fine.

Double check your THHN-2 conductors to make sure they are also listed as THWN. I know it doesn't rain much in Arizona, but by definition, an underground conduit run is a wet location.

There is a very good chance that your existing house panel has all the equipment grounds and neutrals (white) terminated on the same bus. You will need to separate the equipment grounds from the neutrals & make sure that the neutral bus is no longer bonded to the sheet metal of the can or to any equipment ground.

Tom

Few things are harder to put up with than the annoyance of a good example.