It seems to me like there is no way to avoid zombies grabbing me. It's kind of annoying in big fights, there doesn't even seem to be any indication that it's going to happen, everything just stops and the zombie jumps on me and we go down to the ground. I can be mid swing, they can even not be that close to me and they'll do like a CoD knife kill jump towards me and take me down to the ground.

It's kind of annoying the shit out of me.

You can see they are about to do it, typically they extend both arms and do a swing/lunge sort of thing. The problem is in big fights you could be facing 10+ zombies at once so even if you hit like 4 of them the others can still attack you. As you level up it become less and less of a concern because you gain skills like instant escape that let you break grapples immediately, and more mobility moves like the option to shoulder block your way through large groups.

Best thing to do is just never fight a big clump of biters, take on 3-5 at a time tops. Don't do "power" attacks, just regular swings. And if there is more than 5 go with staying at range and using molotovs or other throwables.

If you absolutely have to be in a big group, the whirlwind skill does a pretty nice job of clearing some space. You need to be using a 2 handed weapon for it, though.

There are a few abilities that make it so you can essentially counter the grabbing, though my favorite is the vault, which is tremendously useful for getting through and over zombies quickly. Also, molotovs are your best buddy.

There is an instant escape ability, which, if you push the button prompt quickly enough during the initial grab animation, will push them away without you taking any damage. Really, the best solution is just not to get in a big group to begin with. Better yet, jump up on a ledge and just toss molotovs. That was my strategy for about 90% of the fights.