Note: They are called "collar ties" or ceiling joists, not chords. No biggie.
From the sound of it, I would definitely rebuild the roof structure. The only way you can have 8' spans for a roof is if you have 2.25" - 3" t&g decking instead of 1x sheathing. This decking is only ever used with large beam rafters to support the decking weight.
You need to have rafters every 16" or at most, every 2'. You need to install new rafters and make sure that they are deep enough to fit in a decent amount of insulation UNDER styrofoam "vent keepers" that will keep the sheathing ventilated. This also requires soffit and ridge venting. 2x10 min., better 2x12 or wood Ibeam equiv.. If you can see roofing nails coming through the 1x roof sheathing then it is likely that the integrity of your roofing shingles will not survive the rafter installation. You will pop those nails up and cause leaks by muscle-ing those new rafters in place. You should also be sure to use adequate hurricane (simson) ties to hold everything together, at least at the wall top plate.
The question of the collar ties is a whole other issue. If you just left them hanging out in space I don't think they would look very good. As long as they aren't holding up a ceiling, you can generally remove at least every other one and the structure will still hold together (if they are well attached). The average addition, up to probably 16-20 feet long, probably only needs two collar "beams" spaced at 1/3 intervals, as long as the gable end(s) are strong. These collar beams would have to be a double 2x8 or better and be attached with the proper simpson ties etc. Mind you, this is just ballpark stuff. You might want to confirm this with an architect, engineer or building inspector (you probably need a building permit if you don't have one already). There is nothing wrong with just leaving the collar ties there, other than asthetics (sp?). To remove them altogether you would have to install a ridge beam and end posts that had decent bearing all the way through to a footer.
Overall, there is nothing about the roof in its present condition that couldn't be fixed with 1/4 stick of dynamite.