If a thorough flushing did not remove the debris and the faucets still drip, it will be necessary to disassemble the faucets and check the seat washers. First, turn off all water sources and relieve the pressure at the lavatory faucets. Remove the Hot and Cold handles. There is usually a single screw located below each button on top of each handle. Once the handles are off, carefully remove the bonnet lock and washer. Some will just lift off and others may pop off by gently using a flat blade screwdriver. Remove each stem and bonnet assembly using an adjustable wrench or a deep-well socket. I believe you’ll find them to require a 3/4-inch socket. They are removed by turning counter-clockwise. Now’s the time to carefully inspect each assembly as well as inside the body of the faucet for loose debris and/or mineral build-up. Brush or scrape away any residue you find. Rinse all the parts in clean water. Carefully inspect the bottom seat washer and replace it if it’s dried out, cracked, pinched or distorted in any way. A single screw holds this washer in place. For a quick test (or even a temporary fix), flip that washer over and reassemble the assembly, turn the water back on and check for drips. If the dripping goes away, you’ll know it’s the washer that needs replacing. If the faucet continues to drip periodically, you’ll have to dig further and inspect the seats inside the body of the faucet.
Some faucets have removable seats while others do not. If you see a hexagonal opening in the seats, those can be removed and replaced. Check the top surface of each seat to be sure there are no nicks or cuts. Replace the seats if you feel any sense of scratchiness or roughness. Plastic seats are more prone to damage than brass seats, so check carefully if yours are plastic.
If the faucets, however, contain washerless cartridges they will also contain an O-ring instead of a flat seat washer. It will also have a filtering device in the body of the shank. Be sure to check the filter screen for debris before reassembling. Faucet seats, washers and O-rings are readily available at any hardware store. Take care whenever threading the seats and the stem/bonnet assemblies into the body of the faucet during reassembly. It’s quite easy to cross-thread them! Instead of a 50-cent repair, you’ll be looking at a complete faucet replacement! After reassembly, turn the water pressure back on and check your work. Faucet repair is not as daunting as some might suspect.