I have a 1997 Dodge Dakota with a limited slip 3.55 gear rear differential.I cannot find one with the same gear. Can I replace it with a 3.92 limited slip differential without causing any damage.

If by "rear end" you are referring to the differential housing, many vehicles have a vent hose that allows venting of the differential housing. If the hose comes out of the upper area of the differential and is routed downward, this is most likely what you are seeing.

II think I would try some Motorcraft XL-3 Friction Modifier. That is what Ford uses to fix the same problem in their Limited Slip differentials. You can find it online or at any parts house that carries Motorcraft.

Technically you do not have 4 wheel drive, you have two wheel drive (possibly three-wheel drive). When you engage 4WD you direct power to the front and rear differentials. The front differential is an open style which means that only one wheel will drive, and the rear differential is likely the same type which means that you will only have two wheels driving the vehicle (one front, one rear). If the rear differential is a "posi-trac" or "limited-slip", then both axles should lock and drive the vehicle (hence, three-wheel drive), but this style differential is not as common as it once was.

Changing the differential is not an easy job especially if you've never done one before. Guessing you are keeping your same ring/pinion.
You have to set the backlash with a dial indicator. The pinion has to be meshed with the ring gear at the proper depth. It's a pretty involved process.
See if there's a 4x4 shop near you. They may be more friendly to your ideas. Mudders swap differentials all the time.
Here's a video -

Differential Locker Removal and Open Limited Slip Installed Dodge Ram 2500

000 DODGE TRUCKS DURANGO 5.9L 8-cyl Engine Code [Z]
Limited Slip Differential, Rear 8.25-9.25 W/ TT......GL-5* All TEMPS......75W-140
Standard Differential, Rear 8.25-9.25 W/ TT......GL-5 All TEMPS......75W-140
Limited Slip Differential, Rear 8.25-9.25 W/O TT......GL-5* All TEMPS......75W-90
Standard Differential, Rear 8.25-9.25 W/O TT......GL-5 All TEMPS......75W-90
Note:Trak-Lok axles require Mopar limited slip additive limited slip additive 04318060 or equivalent
Differential, With 7.25 RingGear Rear..........1.4 liters Fill no higher then 3/4 inch below plug hole.
Differential, With 8.25 RingGear Rear..........2.1 liters Fill no higher then 1/4 inch below plug hole.
Differential, With 9.25 RingGear Rear..........2.3 liters Fill no higher then 5/8 inch below plug hole.
Thank you for using fixya and good luck

Jack up the rear, put jack stands at the frame on both sides. Put jack under center of the differential and loosen and remove the shakles, you may need to lower or raise the differential.

Mopart Synthetic Gear Lubricant 75W-90
NOTE:
Trac-Lok T differentials require Limited Slip Additive in the lubricant.
Trac-Rite differentials DO NOT require Limited Slip Additive.

probably something like the star adjuster came loose inside the brake linings...have you checked the back brakes? and if this is a four wheel drive it might be cv joints.

Gear lube is SAE 75W-90 GL-5 gear lubricant. If you have a limited slip diff, add 4 oz of limited slip additive.
Torque cover to 15-20 ft-lbs.

if you have limited slip, make sure you have fresh fluid and limited slip additive. does the differential whine or make noise? if so thats a dead giveaway of a bearing or gear problem, look for metal debris or chunks when you pull the diff cover off