I refuse to cook with anything but charcoal. The best way to do this is to heat your fresh coals separately and then add them. I kept my old Weber when I got my big grill just to have a place to heat up new coals. I just sit my chimney starter on top of the grate and heat them up. When they're ready (gray), I add them using a fireplace shovel or tongs. I have a door in the front of my new grill for adding new coals though. It sounds like you have a weber based on the part about the handles coming up from the grate. If you do have a round Weber, you can actually buy grates that have hinges for lifting up part of the grate to add coals when doing indirect grilling. They also make inserts to keep the coals on the side of the grill if you're using a weber.
I can't stress enough though that you should use lump charcoal instead of briquettes. You won't get that smell you're talking about and you'll find the cooking much better (higher heat, longer burns, etc...). Here in NC, we can get the Full Circle brand at Lowe's Foods which gets really good reviews. See if you can find that brand where you are but if not, here is a link to a review site for lump charcoal.
http://www.nakedwhiz.com/lumpindexpage.htm?bag