Zero drop and "barefoot" shoes are two different things. I first tried minimalist/barefoot style shoes after not liking many of the traditional running shoes I tried. I stuck with these for a few months, but the constant soreness and worry about stepping on something that would hurt my foot got me to switch to some zero drop Altras.
Initially I tried to stay with some of their most minimal shoes, like the escalante. I finally landed at the Torin and really liked it for a while. It worked pretty well for me in the twenty to thirty mile per week range, but as others have mentioned I also suffered a constant stress in my Achilles. Not pain to the point of injury, but just feeling like I was on the edge of an over use injury.
I had the chance to try some max cushion Altras with a 4mm drop and absolutely loved them. My Achilles soreness has gone away completely, and several other minor aches have also improved. I've even pushed my mileage up in recent weeks to a new high of fifty miles. Despite this, I feel better than when I was running 20ish mile weeks with the torins.
I think one major point that many people miss is that even if you're barefoot or wearing zero drop shoes for everything else (walking, work, etc), running is much harder on your feet, legs, and body.
Try a few things and see what you like. Just be aware that some of the barefoot running communities can be kind of dogmatic. Any mention of issues with barefoot shoes is met with the suggestion that you did something wrong (too much too soon, landing on the wrong part of your foot, not paying enough attention for tiny debris on the road).
Good luck finding what works for you! Don't be afraid to change it up. What works today might not be quite right in a few months or years.