Before doing something as hasty as replacing the entire sink, see if you can replace the bathroom faucet instead. You’ll need to know which type of faucet you already own when shopping for a new one. The most common faucet types are the single hole, the 4” triple hole and the 8” triple hole. Unless you want to install a new kind of faucet, keep track of the old type for replacement. If not, check to see if your bathroom vanity is compatible with the faucet type you have in mind.
First, turn off the water supply to the faucet. The valve should be under the sink. If not, shut off the main water valve to the house. Turn on the faucet afterwards to drain water and relieve pressure. Next, disconnect the lift rod from the pop-up assembly and the supply lines from the faucet. Then back off the faucet mounting nuts using your hands or a basin wrench.
If you need to remove the P-trap to have more room to work, disconnect the P-trap, placing a bucket underneath to catch water. Unscrew the slip nuts and washer from the tailpiece and clean the faucet holes. Now you can begin to install the bathroom faucet. As usual, the manufacturer will have specific installation instructions. This will probably involve installing the gasket below the faucet, using putty or sealant. Insert the faucet through the holes for mounting, tightening the mounting nuts on the bottom. After the lift rod is installed, reconnect the water supply lines, using a basin wrench if necessary. Then, remove the aerator. This flushes the faucet of anything that could be blocking it, such as debris from the manufacturing process or sediment.
If not already assembled, attach the handles to the faucet. This is as simple as setting the handle on the base and tightening the setscrew/handle screw that holds it in place.
Reattach the p-trap and trap arm making sure slip nut washers are installed, then re tighten the slip nuts.
Make sure to keep all of the tools the manufacturer provided, as they may be necessary for repairs or maintenance in the future. Test both the hot and cold water, and check the drainpipes and faucets for leaks, tightening as needed. If you find any leaks, check to see if you followed the manufacturer’s instructions with complete accuracy. If not, you may have to work backwards to see what you missed.