There were grinds in the cups in Scotland too and it didn't bother me but I didn't know what they used and if it'd be way worse
I mean, usually it’s just a pinch for me, but if you’re that worried about it, most grocery stores have a machine where you can grind your own beans and choose a courser grind.
Also, my French press is nothing special. I think it was like $20 at Kohls. Not a top end one at all. It doesn’t make a ton of coffee, but it usually gets me two or three cups.