Next time you change your diff fluid, pop the cover off the differential and take a look in there. You could have one of the following:
A) a spool. Seems unlikely, but it could be there. The carrier will look kind of funky because there are no spider gears.
B) an auto locker. This seems most likely to me. This would be a whole carrier replacement that locks up under power. A detroit locker is an auto.
C) lunch box locker. It will have the standard carrier, but the spider gears inside have been replaced with a locking mechanism. Powertrax, lockrite, aussie, spartan; all lunchbox-style lockers.
D) a selectable. This doesn't seem too likely with the symptoms you described. With a selectable, there would be a switch or lever in the cab to activate the locker. Arb, ox, eaton; all selectables.
With an auto locker in the rear diff, you'll be ok on the street, though it may lock or unlock at weird times, might give you a jolt. Also, the tires will chirp as you described, resulting in accelerated tire wear.