We both became eligible at the same time. She got her first dose yesterday, mine is on saturday.
Both were through the cvs website. There is a way to game it and avoid several steps, or at least get them out of the way so you can jump on a date that pops up.
If you just go to their appointment site, you have to select your state. If that state has no appointments, you can't go any further. Once there is one, you can try to make an appointment, but you firsat have to go through a couple of screens of questions, which slows you down. 99% of the time, the slots will be gone by the time you get there. That page refreshes every 15 minutes or so.
But once you get past that initial screen and get to the screen where you can search a zip or city/town name, you can just keep trying. The first frustration is that you will see appts pop up, but when you click on select date, it is gone. These disappear in seconds, and often don't even appear on the intro screen.
If you are lucky and get past the appt location and date, you then need to schedule your second appt. I missed about 5 opp's because there was no second appt available. You can't select just the first, you have to get both.
But I was finally able to grab one at about 1:30 in the morning (couldn't sleep).
There is a bypass though. Check other states (like Texas) that have plenty of appts available. You will quickly get to a screen where you select your state and then you can search zips as above. No waiting for a slot to appear on the intro page.
I called it the texas two step. 
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