I think you are experiencing cluster headaches. Cluster headaches are thought to be a neurovascular headache phenomenon which share features with migraine headaches. The intermittent, severe, unilateral eye pain with intermittent redness and tearing associated with the headaches are fairly classic. They also tend to happen more towards the end of your sleep cycle associated with REM sleep.
Alcohol and tobacco have been known to trigger attacks as has extreme temperatures, watching TV, glare, seasonal allergies, and ironically having sex. For many persons, though, there are no triggers and they get them because they get them.
I would recommend having your doctor send you to see a neurologist. Because cluster headaches (as well as migraines) can be mimickers of serious neurological disease, and also because neurologists are the best trained to treat these, I think having an evaluation by a neurologist is important. It sounds as though it has affected your life enough already.
There are typically two categories of treatments available: prophylactic medicines to take everyday and abortive medicines to take once you feel like it is coming on.
Does this help shed a little more light on things?
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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 see an eye doctor.