many here will tell you to rest, not attempt to exercise during an extended fast. i know all this, but i have a hard time believing others know my body better than i do. if your body allows you to jog, more power to you.
that said, i will be very surprised if you actually have the energy to jog, let alone run, on an extended fast. or if you do, i would not be surprised that you are wiped out for anything else your day may bring.
your body is actually exerting a LOT of energy and reserves to get you through your fast. walking can aid this, by keeping your blood flowing and organs functioning. running may be counterproductive.
good luck, either way! congrats on getting to day 10.
oh- actual advice for not fainting- if you feel lightheaded at all, bend or crouch down lower to the ground until it passes. it wouldn't be the running, per se, that makes you faint. it would be the quick change of position from sitting to standing or the like. change position slowly.