Is There Anything I Can Do to Fix my Crooked Smile?

We are all asymmetric if you look closely enough. The question is whether the asymmetry is enough to warrant correction, and if so what would it take to correct that.
A careful evaluation of your face would include an analysis of the muscles, but of the bone as well. We usually start with a thin cut 3-D CT scan. This shows where there are significant volume differences and significant differences in the bone. Muscle moves different on one side from another if the muscle is spannind bones with differences in projection.
Then the tough choices start about how best to achieve the maximal amount of reasonable symmetry.

You can improve the symmetry of the length of the lip with an upper lip lift and the corner of the mouth with a mid face lift with or without a tendon suspension (depends on how much muscle function you have). You may also benefit from some augmentation of the cheek and deep facial fat as this has likely atrophied from the injury, especially if you had a fracture. And finally, something like Invisalign could level the plane of the teeth to the height of the lip to help improve symmetry when you smile. 

Yes! We can use a tendon to reposition the upper lip into a more symetrical location. This would be an outpatient procedure with excellent results. 

I agree with both Dr's Aldea's and Rand's statements. Your face is asymmetric and you could achieve better symmetry by relaxing the upper lip muscles on the right side of your face. Prior to doing so, however, you should determine if you prefer symmetry with decreased upper lip movement on both sides of your upper lip. Essentially this treatment will lower your normally active right side because there are no treatments that can restore or increase activity of the muscles on the left side of the upper lip where the problem really exists.

The asymmetry with smiling is due a nerve injury weakening the muscles that elevate the lip on the injured side. To correct this, Botox can be  injected into the "stronger" side so that it can "match" the weakened side. Botox in the lower face is an "advanced technique" so find a physician that is familiar with all the facial muscles, such as a Plastic Surgeon.

I would suggest visiting with a facial plastic surgeon. It does appear that you have some facial nerve weakness but without examination of the motion of your face, I can not be sure. I would also want to perform a CT scan to determine if you had a fracture of the underlying bone which did not heal correctly and surgical implants may help. As the other surgeons have pointed out, weakening the other side is also an option.

The only alternative that could potentially help would be Botox placed on the strong side to weaken the strong side to better match with the partially paralyzed side for facial animation.

Ryder,
From the look of the photo you have sustained an injury to the main trunk of your left facial nerve. This has resulted in weakness in the entire left side of your face. Of course the full extent of your deficit can only be assessed with a physical examination. You might benefit from a controlled weakening of the muscles on the right side of your face with Botox to symmetrize your appearance on animation. The alternative is to attempt to increase the animation on the left side of your face with muscle transfer operations. I would recommend a full consultation with a board certified plastic surgeon for a full discussion of your goals, expectations and options, both surgical and non-surgical. Good luck!

Asymmetry is the rule of nature not an exception. In your case you do manifest more muscle activity on the right than on the left.
If you wanted to reduce the amount of upward pull on the Right, see a good Plastic surgeon and have him slightly weaken the LEVATOR A Nasi muscle lifting the right lip upwards.
If you wanted to lift the LEFT corner of the mouth more instead, the Plastic surgeon could weaken the Depressor A oris muscle which opposes upward movement of the corner of the mouth.
I would have this done by the doctor NOT by an injector or an cosmetician.
To fully understand all you need to know about Botox, follow the comprehensive link below -
Dr. P. Aldea

Hi.
You have a nice smile.  But a little Botox  (maybe 5 units) in your right cheek will relax the muscles that pull up the corner of your mouth, and you should have a more symmetrical smile.  If you don't like it, it's gone in 4 months.