Can Botox That is Injected Between Eyebrows Go into Sinuses and Drain Down Throat ?

No this cannot happen.  Botox cannot get into the sinuses when it is injected in between the eyebrows.

Your Botox injections would have to go through all of the soft tissues and then through bone to enter your sinuses. Sinuses do indeed drain into your nasopharynx and eventually throat, but so do tears via the lacrimal system. If you got a bit teary-eyed from your injections, this drains directly into the nose and throat, giving you the sensation you describe.

The same thing happens at a sad part in a movie--that is why people start sniffling: tears down the nose and throat!

Not Botox--tears! Cheers!