I felt the same way as you DLR06 when I was taking my pool classes. I'm a big guy and I hated it when the belt started to slip. A suspenders type harness would certainly help but what I found solved the problem for me was a proper BC.
I bought a jacket style with integrated weights IN MY SIZE. I made sure that I could adjust the waist AND chest straps so the BC felt comfortably snug but not at all constricting. It made a HUGE difference for me. Bouyancy control was much easier because the BC felt like it was part of me rather than something I was struggling against. The integrated weights were heaven-sent. I wear light weights in the back shoulder pockets and heavier up-front. I can also add some more in the front pockets.
Having a proper BC also helps streamline you and that makes a big difference in so many ways.
Finally, when you don't worry about your BC and weights, you'll tend to be less fidgety overall and stillness is the key to good bouyancy control.
In my first couple open water cert dives, I was the Elevator Man. Now, just a few dives later, I can easily ascend and descend with breathe control. I also maintain a much better body position.
I ain't good yet but I sure am BETTER!
When trying on BC's for either purchase or rental, don't be satisfied when you think "I guess this will work" Wait until you say "This feels great!" It will change your whole experience.