I played around with this design too and changed it somewhat and still couldn't get it figured out reliably.
Initially, I added a very small slit to the tubing. Very small is somewhat subjective, and in the beginning it was so small that the solution didn't enter the tubing, and later, when it did, air would bubble out of it, turning the flourish into a foamy mess (I want to dose concentrated stuff).
So, I thought, let's play... with some checkvalves. They should work great, allowing the solution to slowly fill the tubing one way, and when the air comes on, block it so all of it will go through the outflow.
Something unexpected happened, which you saw as well... every 5 seconds or so, a little drop of solution made it through the check valve and added to the dosing volume. That wasn't the problem... the problem was that when the air turned off, some drops would still be stuck in the tubing, now preventing the refill action. I had planned on using that sitting in the stand, going up to the tank a couple of feet. If you have it sitting at tank level, with shorter tubing, it might work better.
To prevent the droplets going through the checkvalve during operation, I tried a couple of things. I reduced the outflow at the end of the tubing, which didn't work well - too small, and the tubing would not refill, too large, droplets would still make their way through the checkvalve.
I added a third checkvalve to increase the backpressure so no liquid would make it through the other valve while the pump was operating. Didn't work well either. Maybe adding the checkvalves complicated things too much, and I need to work on the slit thing. Didn't quite understand the need for it to be covered up and "almost" airtight.
Around that time I received some peristaltic pumps, connected them and put the airpump away for later.