Bearing in mind I'm just learning myself, I thought you might find my experience helpful (or not!)
While I've been learning the sitting trot I was taught to lean back, not massively far, just enough that you have to use your core muscles to keep you in place. The actual lean is probably only a few degrees. I couldn't sit the trot until I did this, and it felt as though I was going to be vaulted forward over the horse's shoulder in both sitting trot and canter.
I've done a fair bit of reading on the topic to help myself, and people do seem to say that leaning forward at all makes it more likely you'll fall off. I know it feels like leaning forward and holding on is the only way to secure your seat, but I found that although shifting my weight back is difficult, I was more secure in this position. Now I just need to work on letting go of the pommel! I strongly suspect that once I'm capable of letting go I should be able to manage the canter, as I couldn't do the rising trot until I stopped holding on.
You mentioned that you were having trouble with the transition from posting trot to canter, perhaps you would find it easier if you begin in sitting trot to start with? That way you are going from sitting to sitting, rather than rising to sitting. I know I find it hard to park my weight deep in the saddle once I've started rising trot. Once you get used to sitting trot > canter you can then practice rising trot > canter.
Is there any way you could use a grab strap, instead of holding the mane? It's probably easier to hang onto, especially with one hand. Is the horse you're riding particularly bouncy? Perhaps you could try on a less bouncy horse?