Dry, Gritty Eyes After Botox

Botox does not have any direct effect on your glands that produce tears, but it does effect the muscle around your eyes ( the orbicularis oculi) and any other muscle that you happen to have injected.
It is important for the doctor to evaluate the position and quality of the lower lid strength before injecting the lateral crow feets and more importantly, the lower lid near the eyelid margin. The effect of the botox is create weakness and drooping of the lower lid--like an ectropion - thus a "dry eye". The good news is that it only lasts as long as Botox does.

Botox itself cannot cause dry eyes, but if it was placed too close to the muscle that supports the lower eyelid, it may cause the lid to droop slightly, which may cause dryness.
This might also explain why the drops are not working. If the lid is drooping, it would not support the drops in the manner it should.

Your eye muscles (orbicularis oculi muscle) help to support your eyelid and keep it from drooping. It is possible that Botox placed too close to this part of the eye muscle can cause decrease in support of the lower eyelid and cause it to droop slightly. If this happens it's feasible to get slightly dry eyes.... This might explain what happened to you.