Re: 02 ford, manual 4.0, transfer case vent tube...
It doesn't actually connect to anything by the engine, just a small vented plastic cap hanging in midair. It should be clipped out of the way of the exhaust but apparently it isn't clipped well enough. The vent cap melted in the same way on my 2003.
To fix the problem permanently, I ran a longer length of reinforced fuel line hose from the t-case and continued it up as high as possible into the engine compartment. I tie wrapped it away from the exhaust along the way and curved it into an inverted "J" bend at the top. The J-bend is tie wrapped to the brake lines at the ABS pump/module with the open outlet facing down. IMO, this is better than the factory setup.
2003 FX4 Level II, Supercharged 5.0L V8, Headers, Duals
BW4406 manual transfer case, Aussie Locker, Torsen L/S, 4.10's
Bilstein 7100 rezzies, Cadillac/Mustang rear discs, Duff traction bars
Keypad Entry, Lock LEDs, ABS switch, sway bar discos
Heated seats, Explorer EATC, Trip Computer and consoles
2003 XLT 2.3L M5OD-R1 reg cab beater, 2011 Taurus SEL, 2016 C-MAX Energi Plug-in Hybrid