Can I quit cold turkey safely on my own?
Who knows?
There is only one way to find out, but I don't recommend it.
Your story sounds familiar, a couple of beers in the morning to alleviate the nasty hangover became typical for me. I also was a successful professional with a ton of responsibility and people counting on me. Publically acknowledging my drinking problem by going to rehab would have been career suicide.
I got medical help with detox a couple of times (slow learner, here) by going to the local "urgent care" medical facility. I was in and out in a coupIe of hours. I was honest and told them I was trying to detox from alcohol, they checked me out, and gave me a prescription for valium which helped with the anxiety, and in my case, the shakes. Within 24 - 36 hours I felt normal.
However, the physical withdrawal is only part of it. Without changing your behavior and the mental/emotional side of things, you might find yourself drunk again in a few weeks.
For me, to get sober (and I'm coming up on 7 years) it took both AA and this website.