Grab the front edge of metal basket and pull it up and down, with force; turn it 90 deg and repeat a few times. It should not appreciably move without the plastic tub outside it moving too. If it clunks up and down you're describing a common bad bearing symptom.

The burning smell is probably the metal basket edge hitting the rubber gasket, if you see a groove wearing in the rubber it's the bearings, you'll see little ground up grey rubber grains often also.

Sometimes the baskets have just become out of round also, nothing you can do about that, it's that same repair as a bearing job, requiring basket and rear tub at a minimum (whole assembly on some newer models).

this will be a rather long post but I'll lay out all of the options I can think of:

I'd like to see a video of it running with sound to be more certain that this dramatic of a repair is necessary but if you are getting free play in that basket it is pretty definitive.

Your model number is incomplete so I don't know exactly which machine you are referring to. Look inside the door for a tag that has model and serial on it so I don't know exactly which machine you are referring to. Look inside the door for a tag that has model and serial on it. If it's a Kenmore the model number will probably start with 110 or 796. Also look at your owner's manual to see if either the outer tub or the metal basket has a longer-term warranty. In general you would need both parts to repair that.

If it is totally out of warranty some people can get away with just replacing the outer tub because it contains the actual bearings but it can be difficult to get the basket shaft out of those bearings if they have been generating a lot of heat.

Most machines only have one year warranties on all the parts now but if it's a few years old you may have longer-term warranties on the tub and or basket..

If not the cost of the job is usually enough to warrant buying a new machine.

As for doing it yourself- It is not a particularly hard job but it is long and somewhat complex so very few people do it themself. However it is possible to do it without any special tools and a friend to help with the heavy lifting. It usually takes us two people and two to three hours to do it. So figure on most of a day if you are doing it for the first time.

If you call Sears about the service they will offer either a home warranty on all your appliances ($50-70 per month) or what is called a service smart agreement on that machine ($220-300). Both allow for a $500 credit towards a new machine if the cost of the repair goes over that number. That repair would usually cost more than that but depending on model you may be able to get that plan to save you a chunk of money and get the repair done. If you like to have a matching washer and dryer and are interested in getting it repaired that is probably a good option.

There is one more less honest approach you could take. If you were to call Sears and ask to buy a protection agreement on that washing machine it would probably cost about $200. That type of plan would cover the full repair. But DO NOT identify that there is an existing problem if you take this route. There would be a short waiting period before you could make a claim so if you can live with the machine for 30 days or so that is an option. I tend toward being overly honest so that feels slimy to me. However I believe that the cost of new appliances versus their lifespan is at a point where if the manufacturer isn't willing to make a decent machine then all bets are off.

Labor rates on this through Sears will be about $300, parts might be $800. If you give your model number you can see at least what the costs at this website would be. Count on Sears service parts costs to be nearly double what you can find online even more than at their own website.