If you can already go days without drinking with no physical consequences you've got a great foundation. Severe alcoholics, like an old housemate I had once, can't do that. Even 24 hours without it is life-threatening for him, he got seizures and had to go to the ER. Alcohol withdrawal is very dangerous. So if you don't go through that your problem with it is not that bad, in terms of substance dependence.
That's the first thing you have to do, is understand the scope of the problem you have without getting emotional about it. You say you feel ashamed. Don't be, because you're not an alcoholic. You just drink more than you would ideally like. Especially since you say you have anxiety issues, don't beat yourself up for a problem that isn't actually that bad.
See if you can't stretch it out. If you can comfortably do three days without, aim for five or six. And then when you do go to buy alcohol get stuff that's high-volume, low-strength. Beer or cider would be ideal. Alcohol is not something you can quit cold turkey, you have to wean yourself off it by drinking low-strength