I've had similar problems before, especially on lesson horses I don't ride often or I'm nervous about (I admittedly play favourites sometimes :3). I'm by no means a pro and you've gotten a ton of really great advice so far, but I've learned some new things lately between reading and switching barns to a new trainer.
As I've been taught with my new trainer, it helps to practice the sitting trot by initially slowing the trot down. Still keeping enough leg on to make sure the horse doesn't break to a walk, but not trying to sit to a full working trot. Deep breaths and remembering to slowly exhale taught me to learn to sink into the tack. After that, I learned to feel how my own hips were moved by the action of the trotting. I sat the trot and while I was going around the ring my instructor asked me to say out loud which of my hips felt like it was being moved forward with the trot. Thinking about that while doing my breathing exercises seemed to help me not tense because I began to imagine that my seat was just another part of the horse and it helped me not to bounce up.
Then, as you go into the transition you want to maintain that same feeling (as everyone else has wisely suggested already!) while giving clear, concise aids. When I was first learning to canter I had a few bad habits. First, my leg aids were not clear and I would get that running trot as well. This was also made worse by leaning too far forward, which made the transition awkward (if it happened at all) because I was too ahead of the horse's centre of gravity. When I stopped leaning I then developed a poor habit of hanging on the reins too tightly, so even if I was giving the correct leg aid my hands were staying "no". In confusion, I would tend to get the running trot again as the poor horse was confused and trying to reconcile the fact that my legs were on and telling him to move. When in doubt, slide your hands up and grab a little mane. I'm not sure about the horse you ride, but the one I learned to canter on my first few times was a big percheron who had a tendency to really throw himself into the first canter stride and do a bit of a rocking horse effect. Grabbing his mane a bit up the neck helped me steady myself (because I didn't have as strong a leg or as much of an independent seat as I do now) without accidentally catching his mouth or giving awkward signals.
As for the leg aid itself I tend to think of my legs as controlling different things. I use my outside leg behind the girth to tell him to canter, but my inside leg giving a squeeze like it's the gas pedal and telling him to move forward. On a corner it helps help you to pick up the correct lead as well, getting the little bend in there. The slide of your outside leg back doesn't have to be a very big move. It shouldn't slide so far back as to upset your balance. The horses I've lessoned on most also tend to be a bit lazy at times, so sometimes the leg aids need to be reinforced with a little tap of the stick behind the saddle to get them moving (or keep them from breaking around the gate or other areas where some of the other students might have let that happen).
My old trainer liked to teach canter transitions out of half seat/two-point, so I'm as new to sitting the trot and then cantering as you are. I'm used to getting into two-point and giving the signal and then either sinking mid-stride or working in a hand gallop. That of course isn't very proper from an equitation standpoint, but he was more concerned with our learning the motion and the fact that a solid two-point wouldn't end up with our getting bounced off at the beginning. We worked more on other things once a canter motion made sense. I don't know if this helps at all or has been redundant with the other great responses, but the point I wanted to mention the most was what your hands were doing. Sometimes when you're bouncing in a sitting trot or getting a running trot when you ask makes you forget about your hands (I know I did all the time!) and my old trainer would constantly nag me about letting go of the horse's head so he could move and that really made all the difference in learning. Grabbing mane at first is good, but eventually as you get your balance your hand will just follow the mouth naturally and be elastic.