What distance and type of terrain do you mostly run? Have you looked at the Trail Talon by Inov8? I am not crazy about the new Trail Roc, but X Talon are great if you are on soft ground.
The Salomon S Lab sense are amazing minimal high performance shoe, but expensive, but worth it. Might be able to find them on sale?
There are not many zero drop options, but Altra is one. They are wide and do not fit me well and tend to be kind of clunky.
For your grass runs, you might really like the Inov8 X Talon line. They are wonderful shoes, very flexible and natural feeling.
The Salomon Sense Ultra are 4mm drop, but a wonderful fit.
I would suggest maybe looking at the Inov8, as you could find 2 shoes, one for road and one for grass - for the price of one Salomon.