New guy, need advice on differencial

[65 P-30]

Hello all,

Another new guy that is in need of advice. What a surprise right.
I'm starting to think that I may have very well bought a pack of trouble with the purchase of a `65 Chevy / Grumman 1 ton step van. I was expecting a lot of work, but things are getting a little out of hand.
The project now is tracking down the source of the rear end howl. I believe the diff is some type of Eaton, but can be mistaken about this. I haven't found any markings as to just what the diff is. If this is any help in identifying the diff, the entire carrier is removable from the front, and the rear of the housing has a removable access cover as well.

Anyway, I've pulled the back cover for a look around. The good news I guess is no parts fell out when the cover was removed. The bad, going by the amount of RTV sealant on the cover, someone had been there before.
After looking things over, there was no obvious damage, such as broken teeth etc.. However, the insides looked more like the inside of an engine that had missed a few oil changes. The parts were black with what look like a light coat of sludge. The oil that was drained was clean. ( my first thought was it was heat related )
The face of the teeth do not appear to have any abnormal wear patterns and have a dull gray finish.
There is what I feel is a lot of freeplay between the ring and pinion. ( the pinion has no contact with the ring gear when moving the pinion from 12:00 almost to the 2:00 position )
So... any ideas of what kind of trouble I'm in here?
Anything that I should check further, or try to adjust?

I'd like to get by without having to remove the carrier for several reasons, but if it needs to come out, it needs to come out.

Thank you for any and all advise.