What's with the long story about the battery? lol
Look at all of the threads regarding 4x4 problems. It could be any number of things. Often times, if there is a lot of snow/ice packed around the transfer case shift motor it will prevent it from shifting. When I use to wheel in Oregon in 18-24"+ of snow I often had to crawl under the truck knock snow out from around the transfer case before it would shift out of 4x4. Get under there, check the connections, and you could also clean the transfer case shift motor for good measure.
2006 F150 FX4 Super Crew - Bilstien 5100s leveled 2", 35" KM2s, 4.56 gears, Detroit locker, Eaton E-Locker, Traction Bars, AEM intake, Xcal 2
2010 VW Golf TDI 6MT