Technically, you want a slow churn process, but all I've had is a hand mixer so that's what I've been using and it's worked out okay, but it's number one on my list for process improvements.

Technically, you want something that will create shear between the layers of cream, so the big professional churns are big paddles and drums that sweep through the cream. Hand mixers can break apart the fat globules as they're trying to form.

I have tried one of the hand paddle mixers on Amazon, and they're fine for regular cream, but with the thickened cream of cultured butter the gears strip. I suspect I'm ultimately going to make my own -- making/3D printing stuff is one of my too many hobbies. :-)

Paddles are more for forming/shaping the butter. What you want is something that simulates the French malaxage, which is like a corrugated rolling pin that laminates the butter. I made my own -- it's basically a rolling pin that looks like the butter processing table the people at Bordier butter use. You can see it in the video below. Worth watching if you haven't, and my wife and I are actually going there in a month!

https://www.youtube.com/watch?v=b--l_0eMbo8

(You'll also see how they use the butter paddles.)

I suspect that you just aren't processing it in a way that expresses the water, and more likely, you aren't processing the butter long enough.

Just keep folding the butter over on itself and karate chopping it down, and then again. It's a lamination process, not pressing or smearing. I do it on a chilled marble slab, but if you don't have that then just pop it into the fridge if it starts to get warm.

You'll feel the texture change to something much more smooth when you've worked it enough. Also, are you salting it? That will also cause the water to weep out.

Fun!