Black tea develops cloudy dregs; how to avoid?
Those particles are bits of tea leaves (etc.) that came out of the tea bags. They're fine enough particles to get through the bags. Basically, tea dust.

You don't notice them at first because they're suspended in the tea. So you could just stir it up before pouring off each cup. Alternatively, disturb the pitcher as minimally as possible, and pour off the tea. Leave the settled tea dust at the bottom.

Rinsing the tea bags in cold water before steeping might help. (Make sure to use cold water so you don't remove much flavor.) 

Other than that, a fine enough filter will remove them. You could try coffee filters, they're cheap enough. Or a nut milk bag. Or a superbag. (Coffee filter is probably the finest filter of those, though by far not the sturdiest).

PS: You might be covering up the oversteeped tea flavor with enough sugar...