I took the truck back to the Ford dealer and had the driver side (left hand side) battery replaced first thing this morning with another Ford service adviser. I feel much better now.
Tinner, it is the Ford dealer where I purchased my truck new in 2012, and I've been getting service discount coupons every year in mail. I was able to get $125 discount on the two batteries and diagnostic services (which resulted in not much more than at local Advance Auto and AutoZone near by). The dealer's been pretty good about covering all previous services on warranty...thus this is the first time I had to pay for my service at the dealer (other than regular filter/oil changes with discount coupons).
Thanks for great advises, Guys! Safe towing!