Will Eye Bags Go Away After Excessive Botox?

Yes, it is most probable that the bags will resolve when the Botox effect dissapates. For the eyes this tends to begin and 3 months and will likely return to normal by 5 months after injection.

I have several patients that experience a prolonged effect from Botox. The longest time span is about 6 months. Relaxation of the muscle of the lower eyelid usually leads to a droopy eyelid. There is a connective tissue barrier, the orbital septum, that keeps the eyefat from bulging. There is a muscular layer between the eyelid skin and the septum and that is what the Botox can relax. It is possible that with relaxation of the muscular sling, you lose that second barrier preventing the fat from bulging. When the Botox effect wears off, the bulging should resolve.
I do not recommend application of Botox along the entire lower eyelid. Wrinkles in that area respond to non-invasive treatments like Thermage and/or Fraxel laser. Surgical option is a blepharoplasty. Make sure you consult with a facial plastic surgeon, plastic surgeon, or oculoplastic surgeon.

Botox actions will fade away in 3-4 months. If the swelling is caused by the needle or by a small bleeder, it should go away much quicker. Best of luck!

Your lower lid appearance will return to the condition it was prior to the Botox injection in a couple of months. There is no permanent effect of the Botox. Some patients with Botox injections around the lower lid have puffiness for a few days to a few weeks. This may be due to the needle injection itself and not the Botox. If the fullness persists longer, then as you mentioned in your question it may be due to relaxation of the muscle.
When outcomes occur that are unexpected you should follow up with your provider.
Assessment of your concerns by an experienced injector should help sort out the cause and minimize the chance of it happening again.