Since he says dampness on the left I'd expect it's not the overfill blowoff. However, the fill plug is on the left there and, even though it's sealed from the factory, I'd get the car up high and look at that first.
I've certainly seen pinion seals go because of misaligned driveshafts, plenty. You can see this most commonly in the jacked-up 4X4 world but it can happen going the other way, too.
The replacement is a mild pita but, you can do this yourself if you're very careful measuring the amount of threads exposed and the position of the pinion nut. Do so, undo the nut, pry the seal out, bang the new one in with a big pipe or socket to spread the load and silicone sealant around it, then return the pinion nut to its exact position. If you get this right, you won't change the pinion depth and have to pull apart the rear end. Had to do this a few times on my own old '88 5.0, waybackwhen.
2014 Premium GT, SGM, Brembo, Auto, Tech, Comfort, etc. GT500s, Steeda UCA, VMP auto tune
2012 GT, Candy Red, Brembo, 3.73, Tech, Comfort, etc. Procal tune, Roush upper, UMI LCAs, Steeda red bracket, GT500s, 28X10X16 Hoosier slicks, best et: 11.91 at 115.23 RIP