Hello and thanks for your question.
You are correct in thinking that your chemotherapy may be causing eye problems. Chemotherapeutic agents are very well-known agents in causing dry eyes. The tearing you are experiencing is likely a reflex tearing in response to the eyes being dry. The fact that your eyelashes are falling out is another sign of tear-film dysfunction which can lead to dry eye.
I would first start out by using artificial tears several times per day. Any kind of artificial tear is okay to use except I would recommend not using the kind that say get-the-red-out or redness-reliever, which I believe Apcon-A is one of them. Use them 4-5x/day to start out with and if you need more then switch to a preservative-free artificial tear which you can use as much as needed. Warm compresses will also help the dry eye get better and you can even use a lubricating ophthalmic ointment 1-2x/day as needed to help the eyes be less dry.
Does this make sense and does that help address your concerns?
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My opinion is solely informative and does not constitute a formal medical opinion or recommendation. For a formal medical opinion and/or recommendation you must be examined by your doctor.