I would go with a LT tire if I were you, even if it involves buying wheels and tires. Currently I have 325/60/18 Nitto Terra Grapplers (load range D) on my F150, but I have 325/60/18 Cooper STTs (load range E) sitting in my garage that I am going to put on as soon as I get the truck out from its yearly winter hibernation. My truck came stock with 275/65/18 Goodyears (load range C) there was a big improvement in handling going to the Terra Grapplers, I don't expect as big of a difference with going to the STTs.
Camping friends have a 2010 Ram 1500, he has dedicated snow tires on his stock 20s and has after market 18s with 275/70/18 (load range E). He is running a small lift, 2.5" front 1.5" rear if I remember correctly.
TT-2013 Passport 3220BH
TV-2004 F150 FX4, not exactly stock...
with 2001 Kodiak K215: 2010-10, 2011-12
with 2012 Grey Wolf 26BH: 2012-19, 2013-24, 2014-11, 2015-6
with 2013 Passport: 2015-13, 2016-15 booked