Hi,
The first thing I'd do is clean the lens and teleconverter contacts. Use a clean rag, lightly dampened with a few drops of isorpropyl alcohol (cheap "rubbing" alcohol works fine), to wipe the contacts on the back of the lens and TC, and the corresponding contacts on the TC and inside the front of the camera. Just be careful to keep off all the optics, camera's mirror, etc.
Next, limit yourself to using the center AF point only. That's the only dual axis/cross type point on the T3i. The rest are single axis and not as sensitive. I don't have a T3i, but my 5DII has similar AF array and I use only the center point most of the time. I'll only use the peripheral AF points with stationary subjects in reasonably good light and contrast. Even then, those peripheral points are either horizontal or vertical... so they will need a roughly perpendicular "edge" or pattern to be able to focus well... a vertical edge with a horizontal AF sensor or a horiz. edge with a vertical AF sensor.
Go out and shoot some more tests with the TC and see if things work better. Be sure to use a tripod, 560mm is a lot of lens to try to get a steady shot, especially on a crop sensor camera (and the 18MP cameras are more susceptible to camera shake than earlier, lower resolution models), even with IS.
Also be sure to use IS, even on a tripod and even when shooting sports.
If panning and using a slower shutter speed to deliberately blur down the background, shift the lens' IS into Mode 2. Otherwise, leave it in Mode 1. IS in Mode 2 is designed specifically for panning shots... In that mode IS only corrects movement on one axis, whatever is perpendicular to the panning direction... So if panning horizontally, only the vertical axis is corrected.
With longer lenses it is simply harder to get the lens on target and keep it focused there. Also, shooting wide open depth of field can be really shallow, focus needs to be even more precise. It might help to stop down the lens a little, say to f4 or f5.6, to give yourself a little extra depth of field, which will give a modest allowance for focus errors.
If subjects are moving, you must be in AI Servo mode. This also will work best with the center AF point only.
If not already doing so, you might want to experiment with Back Button Focusing
, which is a popular focus technique among users of the big white teles, especially for sports/action shooting.
BTW, I'm assuming you got an EF 1.4X II (mark two)... Not an "EX 1.4X VII" ("mark seven"... doesn't exist, the latest is a "III" or "mark III"). In other words, I'm fairly certain sure you mean "version II", not Roman numeral "VII".