I had the same problem, and I've had it since chemo as well.  It might be a blocked tear duct.  If the tears can't exit via the tear duct, they overflow.  My doctor also mentioned the surgery thing, but she said to first try gentle massage.  She showed me how to run my fingers from the inside edge of my eye down along side my nose.  Don't press too hard; you just want to move those tears down the duct that runs along the side of your nose.  The massage worked for me.  I did it several times a day for a few days, and I was fine.  Before she showed me that trick, I thought I just had to put up with it as one more chemo side effect.

Good advice. I had this problem, too, so it's not uncommon. Just to let you know, it does abate after chemo is over. I don't think you want a hole drilled your head - I think I'd rather wait it out! Good luck - PJH