I have a tape that when it plays, it "wobbles", the sound gets all weird, and then the mechanism stops. How do I fix this?

Are you talking about the tape or the tape player?

is it the entire tape or a certain area...?

Is it on certain tapes, or tape player? Tapes do stretch, after being played a certain amount of times, good quailty ones last a bit longer. The other issue is the tape lubrication in the binder becomes dry out or sticky and stick to head and path, that when you get that wobble effect. Other issue is the mylar slip sheets that have gone bad over time inside the cassette causing drag on the take up side, you have to take the cassette apart then remove the slipsheets and spray them with a good dry silicone lubricant and reassemble. You should also spray the dry lube on the rollers and the felt pressure pad.

Cassette tapes life span of all magnetic tape is  10 to 30 years, with a low-quality cassette lasting ten years and a high quality  tape lasting longer. The first indications that a tape is degrading are almost inaudible as signal  is lost to dirt particles.  This loss of sound is called a dropout.  As time  goes on, the damage is more apparent.  Portions of the tape become stretched,  causing the tape to sound as the player was speeding up and slowing down.  Long  shrill, intermittent wheezes and squeals may appear.

Cassette tapes have two or three distinct layers.  From the bottom up, they  include the following: backcoat, omitted altogether in cheap tapes; substrate,  the thickest layer; and topcoat, where the sound is laid down.  The topcoat  consists of magnetic particles embedded in a binder.  Also embedded in the binder are tiny reservoirs of lubricant that keep the tape flexable

Most people attribute the short life cycle of cassette tapes to the binder.  Many of the signs of degradation are caused by defects in the binder. Tapes can  become brittle as lubrication is lost from the binder.  The tape may soft or  become sticky, both of which damage the magnetic signal.  Lastly, excessive  humidity can damage the binder. Reason why you have to be careful how you store them.

The signal you hear when playing a cassette tape is caused by the magnetic  particles.  The magnetic particles are also degrade and distort the signal.  As  the binder stops being cohesive, the magnetic particles literally fall off,  causing dropouts.  In other instances, they lose their magnetic strength  (remanence) which causes loss of the signal. You will see it as brown in color, you have to clean the tape path and demagnetize the heads and metal parts.

First try rewinding and forwarding a few times around 12 times, sometime fixes them, I use a battery drill while cleaning the tape with microfiber cloth. It took multiple passes, it worked for the most part. If not read below and try at your own risk!!

If it's not a important tape throw it away, if it important and you are trying to recover what on the tape then read below. I only done this to reel to reels, not sure how well it will work for a cassette 

If it the blinder tape lubrication that the issue, the best way is to bake it at a low temperature. You can use a hair dryer, or convection ovens . But found that works better is a dehydrator FD50, used it for reel to reels, on that model one the fan on the bottom, so use the top tray. The new model is FD61 use the bottom trays just to be safe (the fan is in the top of the unit). The heat is adjustable, but would check the temperture with a accurate thermometer. Cooking temperature is between 130°F and 140° for reel to reel and bigger tape 1/4 1-4 hours 1/2 2-4 hours 1" inch 3-6 hours 2" 4-8 hours. To accommodate one- and two-inch tapes, modify one tray by cutting out the plastic spokes along the perimeter. This creates a "dummy tray" adding height to the tray below. When baked, the tape will expand and become loose around the hub. For this reason, use flanges to protect the tape from coming apart. For reel to reel, tapes wound on plastic reels with small hubs should be rewound onto large reels with NAB hubs.

http://www.ebay.com/itm/Snackmaster-FD-61-American-Harvest-Food-Dehydrator-NEW-/190736135529?pt=Small_Kitchen_Appliances_US&hash=item2c68c27569

Now if it the tape player that causing the issue, you need to get the unit service, new belts, or idler tire, or both, transport relube and cleaned and some case pinch rollers need to be replace and motor check out for dead spots, sometime issue could be cause by the clutch assy, or incorrect takeup tension. Could also be other issues with the deck, or walkman.

Need to know model of deck, box, or walkman if it is the issue, otherwise pointless, as it could be anything?  

Nice info retrodos.

I always wind the tape to the end and back, which usually work for me.

I try to wind all my tape to the end and back at least one a year if I have not used them in a while, also prevents print through.

It's not the deck, mine's a Technics double deck, hooked up to a headphone jack converter. I'm using it until I can get a nice Walkman. It's on most of the album, both sides. Oh and another thing, I'm not going to stick my Elvis Costello tape in the oven. That's out of the question.

Did the REW/FFD trick, it worked!

Cool