Thanks for the tip! So you see ukulele leash as a provisional solution that can be replaced by firmer grip with your right arm? I'm afraid the right hand grip will be slippy no matter how hard you press the ukulele to your body.
You sound a lot like me when I first started out: You're overthinking and over-worrying.
I can totally relate, though. I was getting really frustrated with the ukulele. I looked at numerous videos and viewed a ton of photos, even bought a couple more books than I needed, always in the hope that one of them would somehow make it really obvious and foolproof how to hold the darn thing.
It all just confused me more, because everyone seemed to hold the ukulele a little differently, some talked about the thumb behind the neck, some seemed to grab the neck like a baseball bat, some recommended to rest the neck in the web between thumb and index finger (at which point I concluded that something must be anatomically wrong with my hands because if I held it like that, I could barely fret the strings at all!), and so on. As a result, I just put the ukulele aside because it made me feel stupid and hopeless, and that wasn't the experience I was looking for.
A bit later, I picked it up again, and I asked here for some help. The answers were essentially: "Get the Uke Leash and don't sweat it, it'll get better". I chewed a bit on that, because I didn't really want to use a strap/leash (another thing to buy, another thing to depend on, etc.) and "it'll get better" seemed so unsatisfying because it was so vague. But I bought a Uke Leash, and wow, what a difference it made!
Sure, I felt like I was using training wheels or those swimming wings that babies wear to prevent them from drowning, but I could actually play chords without cramping up and fumbling around! I did start to swim! It felt good to have the thumb on the neck with the fingers dancing (okay, slowly moving!) across the fretboard, and my progress really sped up. And now, after a little while, with the smaller ukuleles I often "somehow" hold the neck without the Uke Leash doing the holding.
As for the ukulele's body and holding it, that was a similar story for me. I had watched those videos (chiefly one by Cory of HMS) where the players expertly use the lower, inner part of their right arm to gently press the instrument against their body. I tried that and it felt weird, and cramp-y.
I sidestepped the issue by mostly playing while seated in a cross-legged fashion, or on a chair with one leg over the other (and the ukulele sitting on the upper leg), because I felt that I was trying too hard to get it all right, which confused me more. After a while, my body somehow figured out how to best hold the instruments.
It was just like people had told me: it would get better just by playing. It did, once I loosened up and paid less attention to all the technical details, and dropped the fear of acquiring bad habits (I think this is unlikely to happen anyway as long as you put your thumb on the back of the neck).
There, enough rambling from me.
I hope some of this helps, though! I believe it really is as easy as getting the Uke Leash (or another strap) and relaxing. Everything else comes in time.