See, I knew you could do it :) She just needed you to step up and not take her mareish ways.
Developing balance is one of the most important concepts of riding.. since you use it in everything. You being balanced also helps your horse to be balanced, which will in turn help with that inconsistent trot. Since she's a beginner's horse (or so she was labeled) she may not have all the balance that a more advanced horse would with more seasoned and intermediate riders.
You're helping her and she's helping you. That's how it should be. See what a difference respect can make? :)
I'm not sure whether you still ride the horse, but any ways, when a horse goes to the middle of the school or falls through the inside shoulder, just lift your inside hand and rein up a bit, to block the horses shoulder, whilst sending an impulse down the outside rein every few strides. I have found that this works well, just don't overdo it by putting your hand up more that a couple of cm, and don't cross your hand over the horses neck. Hope I helped :)