For the money...the '13 appears a better deal for the cash...similar miles, 3 years newer. I'd personally never another black truck.....but the white truck has a new transmission....which is NOT cheap...think 5-7k...and as you pointed out, no DEF to deal with.
Now the '10 can get better mileage if your willing to delete/tune it...warranty is up so really not much to lose but gain power and mileage. See my sig for my power levels after tune....and i get 18-20 mpg on a flat interstate unloaded.
I'd barter the older truck down..tell them the oil changes mean nothing...make sure the trans is a factory NEW amd not some slimeball local shop (95% don't know how to properly build a dodge 68rfe...seriously, this is not a joke, only two to three builders know it well enough over a factory fresh build that I would trust) and try for 22k or so. Depending on condition of course....
But the '13 had some significant frame, suspension and brake upgrades to consider as well....bit you might be putting in a trans soon too......