You might want to go to Alanon or AA, ask them for advice on how to best deal with your specific case. They've dealt with problems like these for years, and it's far better going in with your eyes open then floundering around with good intentions. If he is as introverted and anxious as you stated, you don't want to isolate him further with dramatics. Nor do you want to keep working as a bandaid solution until it stretches you too far, or just fails to work anymore. At least you can recognize the problem you're actually facing.
You could also look for someone else he, and your family, respects. A friend, or possibly ther priest/spiritual advisor. Someone who can be both objective and considerate to the situation, and be there when he/she is needed. Your cousin is trying to do this to you, but you're clearly uncomfortable with fulfilling this role indefinitely... and you may be too biased at this point.
My father's an alcoholic, it took considerable effort on his part and the support of a local Buddhist community to get him normal.
Perhaps. I'm not convinced that alcohol is the problem. At this point it may just be a coping mechanism to some degree (which is obviously still not a good thing). Professional counseling, though, would probably be beneficial for my family members.
The problem is all of the other pressures that have been building on my uncle. Over the last few years, my uncle's wife has survived a brain tumor, they went through bankruptcy and gave up the house they had been in for 25 years. They moved out into the boondocks (despite my advice that it wasn't a good idea), so now my uncle is driving at least 40 miles to work one way. He works for the Orlando Sentinel, and that company has been squeezing their employees badly for years now, so his job is in a perpetual state of uncertainty. On top of all of this, his daughter has used and abused my aunt & uncle over the years to the point that they are now estranged. Several times before they moved, my uncle was threatened by street thugs in his neighborhood, which led to some legal troubles for a bit until it got straightened out.
Now imagine all of this stuff going on, and then family members who haven't been around at all during any of this suddenly start being judgemental of my uncle during the only time that said family members visit, after more than decade. It's a mess, and I can see several different interpretations to the situation. I think the problem is solvable, but it depends on people letting go of grudges and not being so judgemental, which is easier said than done. Honestly, though, I won't be surprised if it ends in a heart attack.