Can Botox on Crows Feet Cause Muscle Paralysis in Other Areas of my Face?

It is possible that the injections you had for your crow's feet migrated a bit to the muscles in the upper face that control your smile. There are several issues that can affect this.

We usually do not need 20 units for the crows feet. It is typically about 15 for our patients so it's possible a bit more than was necessary was injected.
It could be that the placement of the product was not exactly where it should be and moved to affect other muscles in the area.
Finally, it could be that the migration is in part caused by the fact that the product is overdiluted. The more saline is used with the Botox, the higher the probability that it will migrate to other areas of the face. Different MDs dilute with different amounts of saline and it's worth asking if his/her dilution is higher than is recommended by Allergan (the manufacturer).

First of all, you should only be injected by an experienced Board Certified Plastic Surgeon or Dermatologist.  Second of all, injections for the crow's feet can be very effective, but your injector needs to be very careful not to inject too deep or with too much volume to prevent seeping of the Botox to the levator muscle for the lip.  If this happens, it can affect your smile.  The effect will go away, since Botox is never permanent, but may last for several weeks to months.  

Botox works well but it really is a toxin, and can not be made to wear off extra quick, so too much Botox or placement in the wrong area can really be a problem. Do not take injection lightly, even though it can seem easy when done by an experienced, board-certified dermatologist or plastic surgeon.

It is possible that too much Botox was placed, or that it was placed in the wrong area, if you feel heaviness in your smile and in your cheeks when you were just trying to soften crow's feet. The good news is that it will wear off in 3 months, and you can either correct this result with your next treamment by discussing with the doctor, or find someone else you might be more comfortable with.