Can a Person Become Hypersensitive to Botox?

I would recommend that you stop the Botox treatments. They do not seem to be helping with your original problem - the migraine headaches and you developing a drooping eye two times. I suspect that you are not "sensitive" to Botox, rather, it is more likely a technique problem or you have a preexisting compensated eyelid ptosis.

Because there definitlely seems to be some sort of connection between your Botox injections and the symptoms that you are experiencing, I would suggest refraining from further treatments. Using Botox for medicinal purposes usually requires the higher dosing which may be contributing to the problem. If you would like to continue with Botox for cosmetic purposes, it would be prudent to perform a test first with a small amount to determine if there is a reaction.
Contrary to what the other panel members suggested, there is documentation of those that have developed a hypersensitivity/allergy to the Botox product. In fact, my nurse manager has experienced an immunity that developed into a concerning hypersensitivity. Unfortunately, there is not much interest or financial benefit to provide research on this topic.
Some suggest that dosing frequently and/or in high doses may predispose an individual to an adverse reaction. With our nurse manager, this was most likely the case. A report was filed with Allergan Medical Affairs in hopes of better understanding this phenomenon with no follow up whatsoever on the part of the company. Some have suggested that the sensitvity may be to the albumin used in the product. Regardless, it does exist and should be looked at closely on an individual basis.

Thanks for your question -
I have not seen acquired Botox sensitivity described in the literature. Eyelid drooping after Botox injection can happen in the medication is administered in the wrong area or if you have a significant amount of lid skin excess that your frontalis muscle is working to keep out of your line of site.
You should discuss these issues with your injector. Consider speaking to another board certified dermatologist or plastic surgeon with experience in injecting. And certainly consider other techniques for management of migraines since you have had issues with this type of treatment.
I hope this helps.