There should be a metal tag on the front axle, probably hiding behind the cross member and some grease. There is a stamp on the machined part of the housing where the cover fits on the passenger side. One or two letters.
Depending on wear, it's entirely possible for the differential to spin both front wheels instead of the pinion. I've seen it.
Eagles came with 2.35 and 2.72 axles, what year is yours? Certain combinations were only available some years and that may help us.
Never factory limited slip, although the front is a long-pinion Dana 30 and the back a common AMC 15, both of which have lockers available. All that said, the rear axle was available in some years in other cars with a 2.53 axle...
The 2.35 front will say "20-47" on the tag, the 2.72 will say "18-49".
The 2.35 rear will have an "L" stamped on the axle, the 2.53 will be "K", and the 2.73 "H" or "J".
Yes the front is 18:49 or 2.72 and the rear is 15:41 2.73, or 0.41% different.
Similarly, the front 2.35 axle is 20:47 and the rear is 17:40, or 0.12% different.
I have it all in a spreadsheet.