Attached to one of the diff bolts should be a tag. 2nd row of that tag will start with the gear ratio. If there's an "L" where the decimal point would be, that indicates an LS (so you're looking for 3L73).
Note that when the clutches wear out, it essentially reverts back to an open diff, so I wouldn't be at all surprised if the door sticker is correct and the clutches are just worn out. If you're planning to replace the carrier, it shouldn't matter what is currently installed, because it will come out.
I haven't seen much discussion about swapping the 2nd gen IFS into a 1st gen. I wouldn't be surprised if it's as much work as swapping a solid axle in, but I don't know. You can't just use the gears, because the D35 IFS uses regular cut gears, where the D35 TTB uses reverse cut gears. If you are looking for a set of used gears, you can look for any D35 TTB which was used in '91-'94 Explorers and '91-'97 Rangers (though you have to watch out for the D35/D28 hybrid if you search among Rangers). Because Ford never put an LS in the front from the factory, the only advantage I see to finding a used axle is for the gears. You'll still end up pulling the carrier and installing a new carrier.
Sounds like a good project. Let us know how it goes.