I shoot everything in M now. I learned that lesson a couple of years ago during my grandparents' 60th wedding anniversary. What I shoot mostly is family events/gatherings where the lighting can change dramatically with just a matter of a couple of steps.
Leaving it in Av mode, I could end up going through what you went through... long shutter speeds leading to camera shake and blurred photos. Tv could leave me with some very shallow depth of field.
Usually I take a few "setup" shots before everything starts to get a feel for the area I'll be shooting. Then, after each non-setup photo, I take a quick glance at the Histogram. If I don't like what I see, rather than having the camera choose for me and possibly ending up with a result I really don't like, I can choose to give up some shutter speed. Or if the shutter speed is as low as it can go to shoot handheld, I decide whether to widen the aperture and decrease the depth of field or chance more noise and shoot at a higher ISO.
These quick, on the fly type changes allow me to get the results I want without having to be concerned that the camera might pick the wrong setting to change 
Of course, that all changes when I get into a studio. At a studio, I can set the camera up exactly for the given backdrop and lighting setup. From there, no decision needs to be made. I just snap away... well, until the backdrop or lighting setup changes.