How many miles in between oil changes

Depends on the oil. Full synthetic like Mobil1 or equal they say up to 10,000 miles, I change mine @7500. Conventional. aka mineral oil, 5000 maximum. If you are not sure what is in it now, and are over 5000 change it, and it's perfectly fine to switch from one to another Your Avalon uses 5w30

Noticing that the Avalon oil system takes (and thus has that much more lubricating/cleaning power) 6.5 quarts, not 5 quarts as many other engines do, what does that do to the oil change interval idea?

My XJS took 12 quarts but I still changed it every 3,000 miles. I still recommend 5,000 miles intervals for newer cars.