If you run on surfaces that are clean but manage to step on grass don’t be too worried unless you are a sprinter that believes you shoud be pulling like in Speed Dynamics.
The aqua-sox sound like a good idea. There are also other alternatives at swimming or diving stores that should be pretty cheap. They can double up for use on the beach in shallow water for water resisted drills.
BTW, a client of mine bought an expensive pair of Nike flats here and the bottom tread of one of the shoes fell off in 3 days! Of course, he’d already left town and couldn’t return them. Real craftsmanship there.
How about “reef boots” made for surfing. You can buy 5mm and 7mm thick soles for around £20 (probably $30).
They are pretty stirdy but will make your feet sweat! I ran in 3mm ones on the beach this summer and had no problems. Probably fine on grass with 5-7mm.