Re: Do I need a transmission cooler?
My 2 cents...like many others. Install a tranny cooler and drop your tire size back closer to stock size.
Your tires are too big for your lift (Looks like only a 2" lift up front according to the coil spacers), which is why it rubs when you're turning and flexing. Bad thing if it's rubbing on-road cuz it'll really rub if you go off-road with it. With a lift like that pictured, I'd stick with a 285/75/16 tire max for good clearance
How is the rear of the truck lifted? With blocks, longer shackles, or by lift springs? If it's by blocks, then I'd seriously change things before pulling anything...axle-wrap city! That'll compromise your trailer handling and make things pretty unsafe.
Now to get nit-picky (and maybe a little cruel sounding too). Those aren't mud tires, just a nice semi-aggressive all-terrain tread. They wouldn't do anything if ya did run in the mud...but would be great tires for trail riding out in the rocks and sand. With the tire size & lift you've got, your truck is a pretty solid "mall-crawler" Those tires don't have enough room when you really start working that suspension...impractical for any off-roading and just enough to be a PitA and unsafe for towing anything. Do you do any off-roading at all? And if so, where are the problem spots (tires rubbing, not enough flex, etc)?
You'd be better with a smaller tire size which will also get your final drive ratio back closer to where the engine's powerband can work good enough to pull your boat too. I've pulled a #2400 boat with my little D21 Nissan with no problems (It's lifted 2" and running 31" tires...stock size for that truck). Your truck should have no problem with a #2700 rig behind it if the gearing is right...those tires ARE big enough from stock size to mess with your final drive ratio.
You've got a nice looking truck. If set up right, it should have no problem with your boat.
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BTW, a quick stop at a dealership with the VIN and door sticker info will get ya what axle ratio you have in your truck. You could probably find it online too with a little googling...quite a few Dodge 4x4 forums on the net that would probably have the info too.