A locking rear end or for that matter a working limited slip isn't going to be much of a substitute for 4WD/AWD. About the only time it helps in the snow is if one wheel is still on the pavement. Ford's limited slip only allows for a torque difference of around 30 ft/lbs. between the two rear wheels. What this means is that if both tires are in snow and the one with the least traction starts spinning with only 5 ft/lbs. of torque, the other tire will only get a max. of 35 ft/lbs. If it is in snow, it will still spin just as easily as the other one. So instead of just one tire spinning in the snow, you'll have two. Put in a locker and you can pretty much guarantee that both rear tires will spin in the snow. A locker will also cause the back end to get very loose in low traction situation. With the front wheels pulling via AWD or 4WD it doesn't get nearly as loose. Unfortunately, you only have RWD. There are some circumstances where a locker has its advantages. The situations are when at least one tire has traction. A locker will divide the torque equally among both rear wheels so if you have one tire on a high traction surface and the other in ice, sand or mud, the one tire can still push you along. With an open differential, just the opposite will happen. The tire with the least traction will get all of the torque (just what you don't want to happen). I have driven a RWD car with traction assist, a RWD truck with an open differential, a RWD truck with a Ford limited slip differential and my current 4WD Explorer in the snow and my Explorer by far is the easiest to drive and control. Second on my list would be my RWD Ranger with the open differential or my RWD car with traction assist turned off. The hardest to drive was my RWD truck with the limited slip. The reason being, the open dif. truck and my car with the traction assist turned off would only allow one tire to spin. The other tire stayed planted acting as a sort of anchor if you will.
You can find out if your Explorer came with a limited slip by looking at the
2-digit code listed in the "axle" box on your vehicle certification label inside your driver's door jamb. Post the codes here and someone will tell you what you have. If it starts with "D" then you for sure have a limited slip since a D4 is an LS 3.73 and a D2 is an LS 4.10. There are other codes for limited slips though that start with numbers rather than "D" so it would be best to post your actual codes if in doubt. Also, you can find Ford axle cross-reference tables all over the internet.
If it turns out you do have an open axle, a limited slip would be best if all you want it for is snow covered roads. There are some circumstances where it will help, but there are others where it will hurt so it is a trade-off. You can get one installed for around $300-$500 depending on what you go with and who you pay to put it in. There are several that are designed for the DIY to install, some may require professional installation. I have a Powertrax No-Slip in the rear of my Explorer, however mine is 4WD. It would be overkill on your RWD if all you want it for is snowy roads and definitely would require you to alter your driving style. If you were to go with a locker, your best bet would be a switcheable locker that you can turn off and on when you want. A selectable locker will run $500-$700 installed. I put my Powertrax in myself for about $300.