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I would suggest you ditch AV mode and learn to shoot in Manual mode.
IMO you will have much better results and complete control over your ISO, DOF and Shutter Speed.
Ditto - it took me a while to get used to it, and you'll miss some shots as you do, but I pretty much shoot Manual as my default approach. Also agree with others regarding pushing ISO to keep up your shutter speed, I don't hesitate to shoot above ISO 1000.
Good luck, and keep on shooting