Plumbing – What are the options for fixing a laundry faucet

Sounds like the packing inside of the faucet needs to be replaced, assuming this is similar to an outdoor hose faucet. After you shutoff the water (probably for the whole house if you don't have a separate shutoff for that line) you can open the valve up by unscrewing the nut that the valve stem goes through. You can find new packing material at most home improvement stores that looks like a black string like material, typically right next to all the washers. I'd also suggest replacing the washer at the same time. And clean all the threads and apply fresh plumbers dope before screwing things back together.

If the installing plumber was nice, you can also unscrew the entire faucet from the pipe. Then clean up the threads, put on some fresh pipe dope, and screw on a new faucet. It will cost a few bucks more, but the process is much easier and the result is also nicer. Unfortunately for me, my plumber decided to solder the threads making this task nearly impossible without opening up the wall to cut the pipes.

If you're not comfortable with soldering pipes should things go wrong, and you have to shutoff the water to the entire house, then consider having an appointment with a plumber already scheduled, that you can cancel without incurring any fees, before starting.

You say you replaced the seats. The seat is where the usually rubber washer on the end of the stem seals against the back of the valve body. If you put your finger in the hole where the stem goes you should be able to feel the surface where sealing takes place. This surface can get scratched and allow leaks. In fact, the leak can make the scratch worse over time. Also, debris can become trapped in this space during repair and prevent sealing. So, make sure the seat is not scratched, it can usually be replaced if it is, and make sure no debris is left behind before closing up. Additionally, there is usually a washer made of paper fiber or hard plastic that sits between the valve body and the stem above the threads. If this washer is worn or missing leaking can occur at that point.

Finally, you say you replaced the seats and it got worse. It is possible the seats are not tight or are installed incorrectly, crooked, or cross threaded preventing the washer from seating. Check for that.

Good luck!