Yep, it's your truck. As a 6.0 Ford PowerStroke owner I'll add some things to ponder (not from experience--sarcasm). If you rely on your Trucks alternator to "charge" your Camper batteries you should definitely think about a bigger alternator than stock or OEM. I replaced my OEM alternator five years ago and all my truck electrical issues vanished -- full stop. Since I had plenty of "amperage" available I even added a 6AWG line to charge my batteries separately---I could have never done that with an OEM Alternator. My Leece Neville 230 AMP alternator will charge at idle--you're OEM will struggle if not provide zero charging capability.
Go on-line and invest in a 230AMP alternator - Trust me, You will kick yourself for not doing it a long time ago.
I'd also offer that you need to replace your truck batteries as a pair. Since they are below 12.6V I'd charge them up. I've bought new batteries before and had one go bad - not uncommon. Take your new batteries back to the place you bought them and have them do a "Load" test. You'll know immediately if you have a bad battery and they should replace it for free if you are still in the warranty period. Another word of caution. Just because your Truck battery gets a jump to start does not mean it'll charge. An alternator requires V to excite the alternator. Generators do not. Lots of folks learn this the hard way.
If you can afford it get AGMs for the diesel. Here's a link to a 230AMP. You can search on-line for the best prices. Be advised Leece Neville's are produced in the US. So, there are Chineeesm Knockoffs (go figure). Buy from a reputable dealer.
Leece Neville 230 AMP
