New Wrinkles After Botox - Are They Permanent?

The orbicularis muscle surrounds the eye. Botrox relaxes just the outer fibers, so the other fibers are not affected.
If a patient "tests" their Botox by smiling or "squinching", they may see more activity in fibers that weren't injected.
This should be a very temporary effect.
If it isn't temporary, and that is just the way your eye reacts to the new "balancing act " the muscles are performing, you can always wait for the 3 months and the Botox should go away completely.
22 seems very young to receive Botox injections. While I have no doubt your clinical situation warranted it, I don't usually advise my patients to get Botox for "preventative" reasons; rather only when a problem is actually there.

While it's impossible to know without an evaluation if your lower eyelids actually developed new wrinkles because of the crow's feet injection, what is certain is that if this has happened it will be gone in 3-4 months.
We would suggest going back for a visit to your injector. Dr. Reichner makes a good point that a 1 - 2 week checkup with new patients is appropriate just for this reason. We should know what your results are so that we can address any concerns. If your injector has taken before pictures (he/she should) then check those for evidence that your under eye wrinkles have changed and then address the issue together with your injector.

Sporty
It does sound like your lower eyelid muscles are "over-compensating" for the paralyzed obicularis muscle in the crows feet area. As a Board Certified Surgeon, I feel it is important to see our patients back in the office one to two weeks after their Botox treatment. This allows us to evaluate the effectiveness of the Botox.
Some patients benefit from additional low dose botox injections to correct muscle hyperactivity in other areas. Since your Botox will probably last 3 to 4 months, these notes help us determine what changes need to be made for your next treatment.