How to fix a faucet that drips unless we turn it extremely tightly?
The type of valve that has to be "tightened down" to make a good seal is likely an older valve like the one in this diagram from ThisOldHouse.com:



Over time, there is a washer on the bottom of the stem that wears out.  Forcing the handle tighter can fix the issue, but this ultimately just destroys what's left of the washer even faster.  Normally removing the stem and replacing the flat or cone shaped washer is all you need to do.

Another problem that can occur is the metal "seat" for the stem that stays in the faucet when the stem is removed can become corroded to the point of not providing a good seal.  The seat can be replaced (if you can find one), or they can be ground down with a special tool to provide a new surface for the washer to seal on.  Feel the seat with your finger to see if it's smooth.  Try to just replace the washer first, and only mess with the seat if the drip continues or returns after a short time.

They might be a challenge to find, but you can normally get new seats, an entire stem, or any washer you need to repair the faucet.  A whole new stem will cost on average $15 or so, and they are easy to replace.  You shouldn't need to replace the faucet unless the exterior is in equally bad shape and you just want something new.