Hey everybody, long time reader here, but I finally signed up so I could answer this question.
We had the same issue with well water as we just recently moved to a house in the country, before this we were on city water. The main issue we had was with the sulfur smell (rotten eggs). It appeared as though it had the smell with both hot and cold water, but was definitely worse coming from sink faucets. After speaking with a friend that was a plumber, we learned it had nothing to do with the water, but everything to do with the water heater.
In water heaters there is a long rod inside called an anode rod. It is basically a sacrificial piece of metal that the water destroys, corrodes, and builds deposits on, instead of doing the same to the body of the water heater itself. When the anode rod get covered in all its nastiness you will get a strong sulfur smell. It is typically worse in well water applications.
There are three options to correct this:
1. Replace the anode rod
2. Cut the anode rod off, however this will quickly shorten the water heater tank lifespan
3. Replace the water heater
In our case we replaced the water heater because they are so cheap, $270 for a new electric model. This has solved our sulfur smell problem. So, unless your water heater is brand new you might try one of the above options. You can cut the rod off for free, just to see if that fixes the smell. If it does you should probably then replace the water heater.
I think the reason it seems to smell worse at the sink faucet is because your face normally gets a lot closer to that than a shower faucet.