Sink runs when toilet is flushed.

All the time or just a few drips when water is called for.

As stated, you have to investigate a little more.

Check your water pressure coming into the house. Get a water pressure gauge about 9 dollars at any big box store. hooks up to any hose bib or laundry bib.

If water pressure is over 80 psi install water pressure reducer valve on main water line coming into the house before any fixtures. If you have one already it may need adjusting, cleaning, rebuilding or replacing. Just a suggestion.

When water pressure is that high, you're likely to have banging pipes in the house. If the faucet drips, likely the faucet valves need replacing or the whole thing. If old faucet, I think new faucet with ceramic washers would be better choice. Price-wise, old valve washers (or stems, etc) may cost as much. If nice brass body and chrome or finish still looks good, worth keeping good quality fixtures.