Obrien, from reading your post and considering what others have said it sounds like you airplanes may be set up with the balance point (commonly called center of gravity or CG) a little far back for your level of flying experience. When the stabilization is on, it may be covering up an aft CG condition. Hold the airplane so that the weight of the fuselage is below the wing. Find the point where it balances with the battery in the airplane. There should be a balance point shown in the manual. For stability you will want the airplane to balance at, or just in front of that balance point. This is one of those things that is easy to over do. The saying goes that a nose heavy airplane flies poorly and a tail heavy plane doesn't fly long at all. If you do need to adjust the balance, do it in small increments.
As I type this it also occurs to me that the stability assist may be covering up an issue with over control. Many of us, myself included, as we are learning to fly end up chasing the airplane, responding to its movements with larger than needed movements of the controls. It can be difficult to remember as the airplane is darting around the sky to make smooth small corrections with the sticks. At least some versions of stability assist will limit the movement of the control surfaces no matter how much you move the sticks. As a result you may be overcontrolling and it doesn't show up until you turn off the stability assist.
Speed racer is giving good advice. Even if you have to drive an hour to get there, interaction with a local club and feedback from other fliers can be immensely useful. Just having a set of unoccupied eyeballs on the airplane while you focus on flying can provide a lot of useful feedback.