Can Botox alter facial lymphatic drainage? I have puffy eye bags for 3 months now with no sign of improving.

Hi there! 
This can certainly be a side effect of crows feet botox injections. 
The good news is that this normally settles after the botox completely wears off which is on average 3-4 months. 
If there are any further problems I would suggest follow up with the injector / clinic 
Hope this helps! 
Dr Rupert

Absolutely yes, Botox around the eyes can relax those muscles which normally squeeze lymph fluid out of the area. This is much like if you don't move on a long flight and your leg muscles are inactive, your feet swell.
The good news is that this is dose dependent (so you should be able to have very small amounts of Botox around your eyes, but start VERY small). It is also a temporary effect which should wear off, usually at 3-4 months.

There should be no problem with the 2 procedures but it sounds like you are having injections performed outside of a doctors office and that is not a good idea