If it isn't the drain and/or it's overflow, it might be that the faucet tip is contaminated. You can remove the aerator and place it in a table spoon of bleach in a glass of water for like 20 minutes. Careful to not lose any parts down the drain. Then hold the glass up so you can get the faucet end into the water and hold it there as long as you can and then do it again after a rest. That may stop your odor problem if it isn't caused by H2S in the cold water.
Rinse the aerator well before reinstalling it. If you have a cat, it may be licking the faucet tip.