Well, here is my tale of woe...
I have a 2007 Chrysler Town & Country Limited. The same 3.8L V6 as you do and it is recommended for 5W-20. Since day one, I have used Valvoline conventional 5W-20, and either a Pure One L10241 or PL10241 (whichever was available), OR if they had an OEM MO409, I'd use that.
About 30,000 miles, I started to notice the infamous 3.8L oil consumption. Over the course of two or three oil change cycles it was consistently down to the "Add" line, which is about 1 qt. Brought it to the Chrysler dealer which they noted it, changed the oil and started oil consumption test #1. I followed their letter to the law on what they and Chrysler wanted, and of course it wasn't deemed sufficient and was "within spec" for an engine with the mileage it had on it.
Essentially it took me, 50,000 miles, 3 dealerships, 8 case managers (one included me in a screaming rage on the phone with another customer service rep...), two case numbers and 8 total oil consumption tests to prove that it was burning oil. At the height of the whole situation, within 2,000 miles, it had sucked down 4 quarts of oil on a 5 quart sump. In one of those calls that are recorded for quality assurance, I made the wise crack to the rep that I don't even need to change the oil on this thing, it'll just burn it all by 3k. She found NO amusement in it. Eventually they finally agreed that "it is using more than normal..." and replaced the engine (shortblock). Funny thing, while in for this, I made the comment that they might as well replace the trans too because since day one and multiple reflashes, it's always shifted funny. No, no, no... Sure enough about month later, the trans died... but that's another story!
Anyway, a new shortblock with 14k on it later, all is well. Same diet of Valvoline conventional 5W-20 and PureOne or Mopar filters and it doesn't burn a drop. So, OP, I don't know what else can be done. Just use what they tell you to use and if it burns, it burns. Turns out the oil control rings weren't seated properly once they tore the original block down.
I have made my money's worth with the Lifetime Powertrain warranty I got on this vehicle.